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Dynamic stability and bifurcation point analysis of FG porous core sandwich plate reinforced with graphene platelet

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Abstract

In this study, the dynamic stability and the nonlinear vibration control of a rectangular and symmetric sandwich plate made of functionally graded porous graphene platelet reinforced (FGP-GPL) as a core and two metal face layers under lateral periodic loads are investigated. The FGP-GPL sandwich plate is placed on the Winkler–Pasternak elastic foundation. Three types of distribution along the thickness of the core layer are taken into account to model the porosity of the system. The system's governing equations are derived using the first shear deformation theory (FSDT), the energy method, and Hamilton's principle. The Galerkin method is used next to convert partial differential equations (PDEs) into ordinary differential equations (ODEs). Then, using a multiple-scale method, the equation of motion can be solved. In this mathematical representation, various parameters are considered, such as porosity distribution, porosity coefficient, GPL volume fraction and shape, visco-elastic foundations, geometrical parameters, and mechanical loads, to analyze the dynamic stability and bifurcation status of the sandwich plate.

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Authors and Affiliations

  1. Department of Civil Engineering, K.N. Toosi University of Technology, Tehran, Iran

    Moein Zanjanchi & Saeid Sabouri-Ghomi

  2. Department of Mechanics, Faculty of Engineering, Imam Khomeini International University, Post Box 34149-16818, Qazvin, Iran

    Majid Ghadiri

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  1. Moein Zanjanchi
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Appendices

Appendix 1

$$\begin{aligned} C_{11} & = - \frac{{\pi \left( {32b^{2} A_{11} - 16a^{2} A_{12} + 16a^{2} A_{66} } \right)}}{{36a^{2} b}},\quad C_{12} = - \frac{{\pi \left( {9ab^{2} \pi A_{11} + 9a^{3} \pi A_{66} } \right)}}{{36a^{2} b}}, \\ C_{13} & = - \frac{{\pi \left( {9a^{2} b\pi A_{12} + 9a^{2} b\pi A_{66} } \right)}}{{36a^{2} b}},\quad C_{14} = - \frac{{\pi \left( {9ab^{2} \pi B_{11} + 9a^{3} \pi B_{66} } \right)}}{{36a^{2} b}}, \\ C_{15} & = - \frac{{\pi \left( {9a^{2} b\pi B_{12} + 9a^{2} b\pi B_{66} } \right)}}{{36a^{2} b}}, \\ C_{21} & = - \frac{{\pi \left( { - 16b^{2} A_{12} + 32a^{2} A_{22} + 16b^{2} A_{66} } \right)}}{{36ab^{2} }},\quad C_{22} = - \frac{{\pi \left( {9ab^{2} \pi A_{12} + 9ab^{2} \pi A_{66} } \right)}}{{36ab^{2} }}, \\ C3 & = - \frac{{\pi \left( {9a^{2} b\pi A_{22} + 9b^{3} \pi A_{66} } \right)}}{{36ab^{2} }},\quad C_{24} = - \frac{{\pi \left( {9ab^{2} \pi B_{12} + 9ab^{2} \pi B_{66} } \right)}}{{36ab^{2} }}, \, \\ C_{25} & = - \frac{{\pi \left( {9a^{2} b\pi B_{22} + 9b^{3} \pi B_{66} } \right)}}{{36ab^{2} }}, \, \\ C_{31} & = \frac{{ - 2.283025571109432b^{4} A_{11} + a^{2} \left( { - 4.5660511422b^{2} A_{12} - 2.283025571132a^{2} A_{22} + 3.044034094812b^{2} A_{66} } \right)}}{{a^{3} b^{3} }} \\ C_{32} & = \frac{{5.585053606381853b^{3} A_{11} + 5.585053606381854a^{2} bA_{12} + 2.792526803190927a^{2} bA_{66} }}{{a^{2} b^{2} }} \\ C_{33} & = \frac{{5.585053606381854ab^{2} A_{12} + 5.585053606381853a^{3} A_{22} + 2.792526803190927ab^{2} A_{66} }}{{a^{2} b^{2} }} \\ C_{34} & = \frac{{5.585053606381853b^{3} B_{11} + 5.585053606381854a^{2} bB_{12} + 2.792526803190927a^{2} bB_{66} }}{{a^{2} b^{2} }} \\ C_{35} & = \frac{{5.585053606381854ab^{2} B_{12} + 5.585053606381853a^{3} B_{22} + 2.792526803190927ab^{2} B_{66} }}{{a^{2} b^{2} }} \\ C_{36} & = - 0.7853981633974483bKA_{55} \\ C_{37} & = - 0.7853981633974483aKA_{44} \\ C_{38} & = \frac{1}{{a^{2} b^{2} }}(2.46740110027ab^{3} NTx + 2.4674011002723a^{3} bNTy - 2.4674011002795a^{3} bKA_{44} \\ & \quad - 2.467401100272ab^{3} KA_{55} - 2.467401100272a^{3} bK_{p} - 2.467401100272ab^{3} K_{p} - 0.25a^{3} b^{3} K_{ww} ) \\ \end{aligned}$$
$$\begin{aligned} C_{39} & = 0.25abC_{d} \\ C_{310} & = - 0.25abI_{0} \\ C_{311} & = 2.46740110027ab^{3} N_{d} \\ C_{41} & = - \frac{{32b^{2} \pi B_{11} - 16a^{2} \pi B_{12} + 16a^{2} \pi B_{66} }}{{36a^{2} b}},\quad C_{42} = - \frac{{9ab^{2} \pi^{2} B_{11} + 9a^{3} \pi^{2} B_{66} }}{{36a^{2} b}}, \\ C_{43} & = - \frac{{9a^{2} b\pi^{2} B_{12} + 9a^{2} b\pi^{2} B_{66} }}{{36a^{2} b}} \\ C_{44} & = \frac{{9a^{3} b^{2} KA_{55} + 9ab^{2} \pi^{2} D_{11} + 9a^{3} \pi^{2} D_{66} }}{{36a^{2} b}},\quad C_{45} = - \frac{{9a^{2} b\pi^{2} D_{12} + 9a^{2} b\pi^{2} D_{66} }}{{36a^{2} b}}, \, \\ C_{46} & = - \frac{1}{4}bK\pi A_{55} \\ C_{51} & = \frac{{16b^{2} \pi B_{12} - 32a^{2} \pi B_{22} - 16b^{2} \pi B_{66} }}{{36ab^{2} }},\quad C_{52} = - \frac{{9ab^{2} \pi^{2} B_{12} + 9ab^{2} \pi^{2} B_{66} }}{{36ab^{2} }}, \, \\ C_{53} & = - \frac{{9a^{2} b\pi^{2} B_{22} + 9b^{3} \pi^{2} B_{66} }}{{36ab^{2} }} \\ C_{54} & = - \frac{{9ab^{2} \pi^{2} D_{12} + 9ab^{2} \pi^{2} D_{66} }}{{36ab^{2} }},\quad C_{55} = - \frac{{a^{2} b^{2} KA_{44} + \pi^{2} \left( {a^{2} D_{22} + b^{2} D_{66} } \right)}}{4ab}, \, \\ C_{56} & = - \frac{1}{4}aK\pi A_{44} \\ \end{aligned}$$

Appendix 2

$$\begin{aligned} S_{11} & = \left( \begin{gathered} C_{15} \left( {C_{24} \left( {C_{46} C_{53} - C_{43} C_{56} } \right) + C_{23} \left( { - C_{46} C_{54} + C_{44} C_{56} } \right)} \right) + \hfill \\ C_{14} \left( {C_{25} \left( { - C_{46} C_{53} + C_{43} C_{56} } \right) + C_{23} \left( {C_{46} C_{55} - C_{45} C_{56} } \right)} \right) + \hfill \\ C_{13} \left( {C_{25} \left( {C_{46} C_{54} - C_{44} C_{56} } \right) + C_{24} \left( { - C_{46} C_{55} + C_{45} C_{56} } \right)} \right) \hfill \\ \end{gathered} \right)/S_{1}^{*} \\ S_{12} & = ( - C_{13} C_{25} C_{44} C_{51} + C_{13} C_{24} C_{45} C_{51} + C_{11} C_{25} C_{44} C_{53} \\ & \quad - C_{11} C_{24} C_{45} C_{53} + C_{13} C_{25} C_{41} C_{54} - C_{11} C_{25} C_{43} C_{54} \\ & \quad - C_{13} C_{21} C_{45} C_{54} + C_{11} C_{23} C_{45} C_{54} - C_{13} C_{24} C_{41} C_{55} \\ & \quad + C_{11} C_{24} C_{43} C_{55} + C_{13} C_{21} C_{44} C_{55} \\ & \quad + C_{15} \left( {C_{24} \left( { - C_{43} C_{51} + C_{41} C_{53} } \right) + C_{23} \left( {C_{44} C_{51} - C_{41} C_{54} } \right)} \right. \\ & \quad \left. { + C_{21} \left( { - C_{44} C_{53} + C_{43} C_{54} } \right)} \right) - C_{11} C_{23} C_{44} C_{55} \\ & \quad + C_{14} \left( {C_{25} \left( {C_{43} C_{51} - C_{41} C_{53} } \right) + C_{23} \left( { - C_{45} C_{51} + C_{41} C_{55} } \right)} \right. \\ & \quad \left. {\left. { + C_{21} \left( {C_{45} C_{53} - C_{43} C_{55} } \right)} \right)} \right)/S_{1}^{*} \\ S_{1}^{*} & = \left( {\left( {\left( {C_{15} C_{24} - C_{14} C_{25} } \right)\left( {C_{15} C_{43} - C_{13} C_{45} } \right)} \right.} \right. \\ & \quad \left. { - \left( {C_{15} C_{23} - C_{13} C_{25} } \right)\left( {C_{15} C_{44} - C_{14} C_{45} } \right)} \right) \\ & \quad \times \left( {\left( {C_{15} C_{24} - C_{14} C_{25} } \right)\left( {C_{15} C_{52} - C_{12} C_{55} } \right)} \right. \\ & \quad \left. { - \left( {C_{15} C_{22} - C_{12} C_{25} } \right)\left( {C_{15} C_{54} - C_{14} C_{55} } \right)} \right) \\ & \quad - \left( {\left( {C_{15} C_{24} - C_{14} C_{25} } \right)\left( {C_{15} C_{42} - C_{12} C_{45} } \right)} \right. \\ & \quad \left. { - \left( {C_{15} C_{22} - C_{12} C_{25} } \right)\left( {C_{15} C_{44} - C_{14} C_{45} } \right)} \right) \\ & \quad \times \left( {\left( {C_{15} C_{24} - C_{14} C_{25} } \right)\left( {C_{15} C_{53} - C_{13} C_{55} } \right)} \right. \\ & \quad \left. {\left. { - \left( {C_{15} C_{23} - C_{13} C_{25} } \right)\left( {C_{15} C_{54} - C_{14} C_{55} } \right)} \right)} \right) \\ \end{aligned}$$
$$\begin{aligned} S_{21} & = \left( \begin{gathered} C_{15} \left( {C_{24} \left( { - C_{46} C_{52} + C_{42} C_{56} } \right) + C_{22} \left( {C_{46} C_{54} - C_{44} C_{56} } \right)} \right) \hfill \\ + C_{12} \left( {C_{25} \left( { - C_{46} C_{54} + C_{44} C_{56} } \right) + C_{24} \left( {C_{46} C_{55} - C_{45} C_{56} } \right)} \right) \hfill \\ + C_{14} \left( {C_{25} \left( {C_{46} C_{52} - C_{42} C_{56} } \right) + C_{22} \left( { - C_{46} C_{55} + C_{45} C_{56} } \right)} \right) \hfill \\ \end{gathered} \right)/S_{2}^{*} \\ S_{22} & = (C_{12} C_{25} C_{44} C_{51} - C_{12} C_{24} C_{45} C_{51} - C_{11} C_{25} C_{44} C_{52} \\ & \quad + C_{11} C_{24} C_{45} C_{52} - C_{12} C_{25} C_{41} C_{54} + C_{11} C_{25} C_{42} C_{54} \\ & \quad + C_{12} C_{21} C_{45} C_{54} + C_{15} \left( {C_{24} \left( {C_{42} C_{51} - C_{41} C_{52} } \right)} \right. \\ & \quad \left. { + C_{22} \left( { - C_{44} C_{51} + C_{41} C_{54} } \right) + C_{21} \left( {C_{44} C_{52} - C_{42} C_{54} } \right)} \right) \\ & \quad + C_{12} C_{24} C_{41} C_{55} - C_{11} C_{24} C_{42} C_{55} - C_{12} C_{21} C_{44} C_{55} \\ & \quad + C_{11} C_{22} C_{44} C_{55} - C_{11} C_{22} C_{45} C_{54} \\ & \quad + C_{14} \left( {C_{25} \left( { - C_{42} C_{51} + C_{41} C_{52} } \right) + C_{22} \left( {C_{45} C_{51} - C_{41} C_{55} } \right)} \right. \\ & \quad \left. {\left. { + C_{21} \left( { - C_{45} C_{52} + C_{42} C_{55} } \right)} \right)} \right)/S_{2}^{*} \\ S_{2}^{*} & = (C_{15} C_{24} C_{43} C_{52} - C_{14} C_{25} C_{43} C_{52} - C_{15} C_{23} C_{44} C_{52} \\ & \quad + C_{13} C_{25} C_{44} C_{52} + C_{14} C_{23} C_{45} C_{52} - C_{13} C_{24} C_{45} C_{52} \\ & \quad - C_{15} C_{24} C_{42} C_{53} + C_{14} C_{25} C_{42} C_{53} + C_{15} C_{22} C_{44} C_{53} \\ & \quad - C_{12} C_{25} C_{44} C_{53} - C_{14} C_{22} C_{45} C_{53} + C_{12} C_{24} C_{45} C_{53} \\ & \quad + C_{15} C_{23} C_{42} C_{54} - C_{13} C_{25} C_{42} C_{54} - C_{15} C_{22} C_{43} C_{54} \\ & \quad + C_{12} C_{25} C_{43} C_{54} + C_{13} C_{22} C_{45} C_{54} - C_{12} C_{23} C_{45} C_{54} \\ & \quad - C_{14} C_{23} C_{42} C_{55} + C_{13} C_{24} C_{42} C_{55} + C_{14} C_{22} C_{43} C_{55} \\ & \quad - C_{12} C_{24} C_{43} C_{55} - C_{13} C_{22} C_{44} C_{55} + C_{12} C_{23} C_{44} C_{55} )) \\ \end{aligned}$$
$$\begin{aligned} S_{31} & = \left( \begin{gathered} C_{15} \left( {C_{23} \left( {C_{46} C_{52} - C_{42} C_{56} } \right) + C_{22} \left( { - C_{46} C_{53} + C_{43} C_{56} } \right)} \right) \hfill \\ + C_{13} \left( {C_{25} \left( { - C_{46} C_{52} + C_{42} C_{56} } \right) + C_{22} \left( {C_{46} C_{55} - C_{45} C_{56} } \right)} \right) \hfill \\ + C_{12} \left( {C_{25} \left( {C_{46} C_{53} - C_{43} C_{56} } \right) + C_{23} \left( { - C_{46} C_{55} + C_{45} C_{56} } \right)} \right) \hfill \\ \end{gathered} \right)/S_{3}^{*} \\ S_{32} & = ( - C_{12} C_{25} C_{43} C_{51} + C_{12} C_{23} C_{45} C_{51} + C_{11} C_{25} C_{43} C_{52} \\ & \quad - C_{11} C_{23} C_{45} C_{52} + C_{12} C_{25} C_{41} C_{53} + C_{11} C_{22} C_{45} C_{53} \\ & \quad + C_{15} \left( {C_{23} \left( { - C_{42} C_{51} + C_{41} C_{52} } \right) + C_{22} \left( {C_{43} C_{51} - C_{41} C_{53} } \right) + C_{21} \left( { - C_{43} C_{52} + C_{42} C_{53} } \right)} \right) \\ & \quad - C_{12} C_{23} C_{41} C_{55} + C_{11} C_{23} C_{42} C_{55} + C_{12} C_{21} C_{43} C_{55} \\ & \quad - C_{11} C_{22} C_{43} C_{55} - C_{11} C_{25} C_{42} C_{53} - C_{12} C_{21} C_{45} C_{53} \\ & \quad + C_{13} \left( {C_{25} \left( {C_{42} C_{51} - C_{41} C_{52} } \right) + C_{22} \left( { - C_{45} C_{51} + C_{41} C_{55} } \right)} \right. \\ & \quad \left. {\left. { + C_{21} \left( {C_{45} C_{52} - C_{42} C_{55} } \right)} \right)} \right)/S_{3}^{*} \\ S_{3}^{*} & = (C_{15} C_{24} C_{43} C_{52} - C_{14} C_{25} C_{43} C_{52} - C_{15} C_{23} C_{44} C_{52} \\ & \quad + C_{13} C_{25} C_{44} C_{52} + C_{14} C_{23} C_{45} C_{52} - C_{13} C_{24} C_{45} C_{52} \\ & \quad - C_{15} C_{24} C_{42} C_{53} + C_{14} C_{25} C_{42} C_{53} + C_{15} C_{22} C_{44} C_{53} \\ & \quad - C_{12} C_{25} C_{44} C_{53} - C_{14} C_{22} C_{45} C_{53} + C_{12} C_{24} C_{45} C_{53} \\ & \quad { + }C_{15} C_{23} C_{42} C_{54} - C_{13} C_{25} C_{42} C_{54} - C_{15} C_{22} C_{43} C_{54} \\ & \quad + C_{12} C_{25} C_{43} C_{54} + C_{13} C_{22} C_{45} C_{54} - C_{12} C_{23} C_{45} C_{54} \\ & \quad - C_{14} C_{23} C_{42} C_{55} + C_{13} C_{24} C_{42} C_{55} + C_{14} C_{22} C_{43} C_{55} \\ & \quad - C_{12} C_{24} C_{43} C_{55} - C_{13} C_{22} C_{44} C_{55} + C_{12} C_{23} C_{44} C_{55} )))) \\ \end{aligned}$$
$$\begin{aligned} S_{41} & = \left( \begin{gathered} C_{14} \left( {C_{23} \left( { - C_{46} C_{52} + C_{42} C_{56} } \right) + C_{22} \left( {C_{46} C_{53} - C_{43} C_{56} } \right)} \right) \hfill \\ + C_{12} \left( {C_{24} \left( { - C_{46} C_{53} + C_{43} C_{56} } \right) + C_{23} \left( {C_{46} C_{54} - C_{44} C_{56} } \right)} \right) \hfill \\ + C_{13} \left( {C_{24} \left( {C_{46} C_{52} - C_{42} C_{56} } \right) + C_{22} \left( { - C_{46} C_{54} + C_{44} C_{56} } \right)} \right) \hfill \\ \end{gathered} \right)/S_{4}^{*} \\ S_{42} & = (C_{12} C_{24} C_{43} C_{51} - C_{12} C_{23} C_{44} C_{51} - C_{11} C_{24} C_{43} C_{52} \\ & \quad + C_{11} C_{23} C_{44} C_{52} - C_{12} C_{24} C_{41} C_{53} + C_{11} C_{24} C_{42} C_{53} \\ & \quad + C_{12} C_{21} C_{44} C_{53} + C_{14} \left( {C_{23} \left( {C_{42} C_{51} - C_{41} C_{52} } \right)} \right. \\ & \quad \left. { + C_{22} \left( { - C_{43} C_{51} + C_{41} C_{53} } \right) + C_{21} \left( {C_{43} C_{52} - C_{42} C_{53} } \right)} \right) \\ & \quad + C_{12} C_{23} C_{41} C_{54} - C_{11} C_{23} C_{42} C_{54} - C_{12} C_{21} C_{43} C_{54} \\ & \quad + C_{11} C_{22} C_{43} C_{54} - C_{11} C_{22} C_{44} C_{53} + C_{13} \left( {C_{24} \left( { - C_{42} C_{51} + C_{41} C_{52} } \right)} \right. \\ & \quad \left. {\left. { + C_{22} \left( {C_{44} C_{51} - C_{41} C_{54} } \right) + C_{21} \left( { - C_{44} C_{52} + C_{42} C_{54} } \right)} \right)} \right)/S_{4}^{*} \\ S_{4}^{*} & = (C_{15} C_{24} C_{43} C_{52} - C_{14} C_{25} C_{43} C_{52} - C_{15} C_{23} C_{44} C_{52} \\ & \quad + C_{13} C_{25} C_{44} C_{52} + C_{14} C_{23} C_{45} C_{52} - C_{13} C_{24} C_{45} C_{52} \\ & \quad - C_{15} C_{24} C_{42} C_{53} + C_{14} C_{25} C_{42} C_{53} + C_{15} C_{22} C_{44} C_{53} \\ & \quad - C_{12} C_{25} C_{44} C_{53} - C_{14} C_{22} C_{45} C_{53} + C_{12} C_{24} C_{45} C_{53} \\ & \quad + C_{15} C_{23} C_{42} C_{54} - C_{13} C_{25} C_{42} C_{54} - C_{15} C_{22} C_{43} C_{54} \\ & \quad + C_{12} C_{25} C_{43} C_{54} + C_{13} C_{22} C_{45} C_{54} - C_{12} C_{23} C_{45} C_{54} \\ & \quad - C_{14} C_{23} C_{42} C_{55} + C_{13} C_{24} C_{42} C_{55} + C_{14} C_{22} C_{43} C_{55} \\ & \quad - C_{12} C_{24} C_{43} C_{55} - C_{13} C_{22} C_{44} C_{55} + C_{12} C_{23} C_{44} C_{55} )) \\ \end{aligned}$$

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Zanjanchi, M., Ghadiri, M. & Sabouri-Ghomi, S. Dynamic stability and bifurcation point analysis of FG porous core sandwich plate reinforced with graphene platelet. Acta Mech 234, 5015–5037 (2023). https://doi.org/10.1007/s00707-023-03638-8

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