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Nonlinear low-velocity impact response of graphene platelet-reinforced metal foam cylindrical shells under axial motion with geometrical imperfection

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Abstract

This paper investigates the nonlinear low-velocity impact response of geometrically imperfect graphene platelets-reinforced metal foams (GPLRMF) cylindrical shells under axial motion. Based on Love’s thin shell theory and von Karman's geometric nonlinearity, the nonlinear equations of motion for the GPLRMF cylindrical shells are established. Then, combined with the Galerkin method, the fourth-order Runge–Kutta technique is adopted to simulate the time history curves of the GPLRMF cylindrical shells, in which the simply supported boundary condition is considered. Subsequently, the nonlinear impact force between the GPLRMF cylindrical shells and impactor is obtained by using Hertz contact force model and Newton's second law. In the end, numerical results are presented to investigate the effects of various parameters including foam distribution, graphene platelets (GPLs) distribution pattern, GPLs weight fraction, foam coefficient, axially moving velocity of the cylindrical shells, radius and initial velocity of the spherical impactor, pre-stressing force, damping coefficient and initial geometrical imperfection on the nonlinear low-impact response of the GPLRMF cylindrical shells.

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Acknowledgements

The authors acknowledge this work is supported by the talent introduction project of Chongqing University (02090011044159) and Fundamental Research Funds for the Central Universities (2022CDJXY-005).

Funding

The talent introduction project of Chongqing University, 02090011044159, Gui-Lin She, Fundamental Research Funds for Central Universities of the Central South University, 2022CDJXY-005, Gui-Lin She.

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Authors and Affiliations

  1. College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing, 400044, China

    Yi-Wen Zhang & Gui-Lin She

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  1. Yi-Wen Zhang
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Correspondence to Gui-Lin She.

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Appendix 1

$$ L_{11} = A_{11} \frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{A_{66} }}{{R^{2} }}\frac{{\partial^{2} u}}{{\partial \theta^{2} }} $$
(A1)
$$ L_{12} = \left[ {A_{12} + A_{66} + \frac{1}{R}\left( {B_{12} + 2B_{66} } \right)} \right]\frac{1}{R}\frac{{\partial^{2} v}}{\partial x\partial \theta } $$
(A2)
$$ \begin{aligned} L_{{13}} = & \frac{{A_{{12}} }}{R}\frac{{\partial w}}{{\partial x}} - B_{{11}} \frac{{\partial ^{3} w}}{{\partial x^{3} }} - \frac{{\left( {B_{{12}} + 2B_{{66}} } \right)}}{{R^{2} }}\frac{{\partial ^{3} w}}{{\partial x\partial \theta ^{2} }} \\ & + A_{{11}} \left( {\frac{{\partial ^{2} w}}{{\partial x^{2} }}\frac{{\partial w^{*} }}{{\partial x}} + \frac{{\partial w}}{{\partial x}}\frac{{\partial ^{2} w^{*} }}{{\partial x^{2} }}} \right) + A_{{12}} \left( {\frac{1}{{R^{2} }}\frac{{\partial ^{2} w}}{{\partial x\partial \theta }}\frac{{\partial w^{*} }}{{\partial \theta }} + \frac{1}{{R^{2} }}\frac{{\partial w}}{{\partial \theta }}\frac{{\partial ^{2} w^{*} }}{{\partial x\partial \theta }}} \right) \\ & + \frac{{A_{{66}} }}{{R^{2} }}\left( {\frac{{\partial ^{2} w}}{{\partial x\partial \theta }}\frac{{\partial w^{*} }}{{\partial \theta }} + \frac{{\partial w}}{{\partial \theta }}\frac{{\partial ^{2} w^{*} }}{{\partial x\partial \theta }} + \frac{{\partial ^{2} w}}{{\partial \theta ^{2} }}\frac{{\partial w^{*} }}{{\partial x}} + \frac{{\partial ^{2} w^{*} }}{{\partial \theta ^{2} }}\frac{{\partial w}}{{\partial x}}} \right) \\ \end{aligned} $$
(A3)
$$ P_{1} = A_{11} \frac{\partial w}{{\partial x}}\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{A_{12} }}{{R^{2} }}\frac{\partial w}{{\partial \theta }}\frac{{\partial^{2} w}}{\partial x\partial \theta } + \frac{{A_{66} }}{{R^{2} }}\left( {\frac{{\partial^{2} w}}{{\partial \theta^{2} }}\frac{\partial w}{{\partial x}} + \frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{\partial w}{{\partial \theta }}} \right) $$
(A4)
$$ L_{21} = \left[ {A_{66} + A_{12} + \frac{1}{R}\left( {2B_{66} + B_{12} } \right)} \right]\frac{1}{R}\frac{{\partial^{2} u}}{\partial x\partial \theta } $$
(A5)
$$ \begin{aligned} L_{{22}} = & \left[ {A_{{66}} + \frac{1}{R}\left( {4B_{{66}} + \frac{4}{R}D_{{66}} } \right)} \right]\frac{{\partial ^{2} v}}{{\partial x^{2} }} \\ & + \left[ {A_{{22}} + \frac{1}{R}\left( {2B_{{22}} + \frac{1}{R}D_{{22}} } \right)} \right]\frac{1}{{R^{2} }}\frac{{\partial ^{2} v}}{{\partial \theta ^{2} }} \\ \end{aligned} $$
(A6)
$$ \begin{gathered} L_{23} = - \left[ {2B_{66} + B_{12} + \frac{1}{R}\left( {4D_{66} + D_{12} } \right)} \right]\frac{1}{R}\frac{{\partial^{3} w}}{{\partial x^{2} \partial \theta }} + \frac{1}{{R^{2} }}\left( {A_{22} + \frac{{B_{22} }}{R}} \right)\frac{\partial w}{{\partial \theta }} - \left( {B_{22} + \frac{1}{R}D_{22} } \right)\frac{1}{{R^{3} }}\frac{{\partial^{3} w}}{{\partial \theta^{3} }} \hfill \\ + \left( {A_{66} + \frac{{B_{66} }}{R}} \right)\frac{1}{R}\left( {\frac{{\partial^{2} w^{*} }}{{\partial x^{2} }}\frac{\partial w}{{\partial \theta }} + \frac{{\partial w^{*} }}{\partial x}\frac{{\partial^{2} w}}{\partial x\partial \theta } + \frac{\partial w}{{\partial x}}\frac{{\partial^{2} w^{*} }}{\partial x\partial \theta } + \frac{{\partial w^{*} }}{\partial \theta }\frac{{\partial^{2} w}}{{\partial x^{2} }}} \right) \hfill \\ + \left( {A_{12} + \frac{{B_{12} }}{R}} \right)\frac{1}{R}\left( {\frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{{\partial w^{*} }}{\partial x} + \frac{\partial w}{{\partial x}}\frac{{\partial^{2} w^{*} }}{\partial x\partial \theta }} \right) + \frac{1}{{R^{3} }}\left( {A_{22} + \frac{{B_{22} }}{R}} \right)\left( {\frac{{\partial^{2} w}}{{\partial \theta^{2} }}\frac{{\partial w^{*} }}{\partial \theta } + \frac{\partial w}{{\partial \theta }}\frac{{\partial^{2} w^{*} }}{{\partial \theta^{2} }}} \right) \hfill \\ \end{gathered} $$
(A7)
$$ P_{2} = \frac{1}{R}\left( {A_{66} + 2\frac{{B_{66} }}{R}} \right)\left( {\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{\partial w}{{\partial \theta }} + \frac{\partial w}{{\partial x}}\frac{{\partial^{2} w}}{\partial x\partial \theta }} \right) + \frac{1}{R}\left( {A_{12} + \frac{{B_{12} }}{R}} \right)\frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{\partial w}{{\partial x}} + \frac{1}{{R^{3} }}\left( {A_{22} + \frac{{B_{22} }}{R}} \right)\frac{{\partial^{2} w}}{{\partial \theta^{2} }}\frac{\partial w}{{\partial \theta }} $$
(A8)
$$ L_{31} = B_{11} \frac{{\partial^{3} u}}{{\partial x^{3} }} + \frac{1}{{R^{2} }}\left( {2B_{66} + B_{12} } \right)\frac{{\partial^{3} u}}{{\partial x\partial \theta^{2} }} - \frac{{A_{12} }}{R}\frac{\partial u}{{\partial x}} $$
(A9)
$$ L_{32} = \frac{1}{R}\left[ {B_{12} + 2B_{66} + \frac{1}{R}\left( {4D_{66} + D_{12} } \right)} \right]\frac{{\partial^{3} v}}{{\partial x^{2} \partial \theta }} + \frac{1}{{R^{3} }}\left( {B_{22} + \frac{{D_{22} }}{R}} \right)\frac{{\partial^{3} v}}{{\partial \theta^{3} }} - \frac{1}{R}\left( {\frac{{A_{22} }}{R} + \frac{{B_{22} }}{{R^{2} }}} \right)\frac{\partial v}{{\partial \theta }} $$
(A10)
$$ \begin{gathered} L_{33} = \frac{{2B_{12} }}{R}\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{2B_{22} }}{{R^{3} }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }} - D_{11} \frac{{\partial^{4} w}}{{\partial x^{4} }} - \frac{{D_{22} }}{{R^{4} }}\frac{{\partial^{4} w}}{{\partial \theta^{4} }} - \frac{{(2D_{12} + 4D_{66} )}}{{R^{2} }}\frac{{\partial^{4} w}}{{\partial x^{2} \partial \theta^{2} }} - \frac{{A_{22} }}{{R^{2} }}w \hfill \\ + B_{11} \left( {\frac{{\partial^{3} w}}{{\partial x^{3} }}\frac{{\partial w^{*} }}{\partial x} + 2\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w^{*} }}{{\partial x^{2} }} + \frac{{\partial^{3} w^{*} }}{{\partial x^{3} }}\frac{\partial w}{{\partial x}}} \right) + \frac{{B_{12} }}{{R^{2} }}\left( {\frac{{\partial^{3} w}}{{\partial x^{2} \partial \theta }}\frac{{\partial w^{*} }}{\partial \theta } + 2\frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{{\partial^{2} w^{*} }}{\partial x\partial \theta } + \frac{\partial w}{{\partial \theta }}\frac{{\partial^{3} w^{*} }}{{\partial x^{2} \partial \theta }}} \right) \hfill \\ + \frac{{2B_{66} }}{{R^{2} }}\left( \begin{gathered} \frac{\partial w}{{\partial \theta }}\frac{{\partial^{3} w^{*} }}{{\partial x^{2} \partial \theta }} + \frac{{\partial^{2} w^{*} }}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }} + 2\frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{{\partial^{2} w^{*} }}{\partial x\partial \theta } \hfill \\ + \frac{{\partial^{3} w}}{{\partial x\partial \theta^{2} }}\frac{{\partial w^{*} }}{\partial x} + \frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w^{*} }}{{\partial \theta^{2} }} + \frac{{\partial w^{*} }}{\partial \theta }\frac{{\partial^{3} w}}{{\partial x^{2} \partial \theta }} + \frac{{\partial^{3} w^{*} }}{{\partial x\partial \theta^{2} }}\frac{\partial w}{{\partial x}} \hfill \\ \end{gathered} \right) \hfill \\ + \frac{{B_{12} }}{{R^{2} }}\left( {\frac{{\partial^{3} w}}{{\partial x\partial \theta^{2} }}\frac{\partial w}{{\partial x}} + 2\frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{{\partial^{2} w^{*} }}{\partial x\partial \theta } + \frac{\partial w}{{\partial x}}\frac{{\partial^{3} w^{*} }}{{\partial x\partial \theta^{2} }}} \right) \hfill \\ + \frac{{B_{22} }}{{R^{4} }}\left( {\frac{{\partial^{3} w}}{{\partial \theta^{3} }}\frac{{\partial w^{*} }}{\partial \theta } + 2\frac{{\partial^{2} w}}{{\partial \theta^{2} }}\frac{{\partial^{2} w^{*} }}{{\partial \theta^{2} }} + \frac{{\partial^{3} w^{*} }}{{\partial \theta^{3} }}\frac{\partial w}{{\partial \theta }}} \right) - \frac{{A_{12} }}{R}\frac{\partial w}{{\partial x}}\frac{{\partial w^{*} }}{\partial x} - \frac{{A_{22} }}{{R^{3} }}\frac{\partial w}{{\partial \theta }}\frac{{\partial w^{*} }}{\partial \theta } \hfill \\ \end{gathered} $$
(A11)
$$ \begin{gathered} P_{3} = B_{11} \left( {\frac{{\partial^{3} w}}{{\partial x^{3} }}\frac{\partial w}{{\partial x}} + \frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial x^{2} }}} \right) + \frac{{B_{12} }}{{R^{2} }}\left( {\frac{{\partial^{3} w}}{{\partial x^{2} \partial \theta }}\frac{\partial w}{{\partial \theta }} + \frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{{\partial^{2} w}}{\partial x\partial \theta }} \right) + \frac{{A_{22} }}{{R^{4} }}\frac{\partial w}{{\partial \theta }}\frac{{\partial w^{*} }}{\partial \theta }\frac{{\partial^{2} w}}{{\partial \theta^{2} }} \hfill \\ + \frac{{2B_{66} }}{{R^{2} }}\left( {\frac{{\partial^{3} w}}{{\partial x^{2} \partial \theta }}\frac{\partial w}{{\partial \theta }} + \frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }} + \frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{{\partial^{2} w}}{\partial x\partial \theta } + \frac{\partial w}{{\partial x}}\frac{{\partial^{3} w}}{{\partial x\partial \theta^{2} }}} \right) + \frac{{B_{12} }}{{R^{2} }}\left( {\frac{{\partial^{3} w}}{{\partial x\partial \theta^{2} }}\frac{\partial w}{{\partial x}} + \frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{{\partial^{2} w}}{\partial x\partial \theta }} \right) \hfill \\ + \frac{{B_{22} }}{{R^{4} }}\left( {\frac{{\partial^{3} w}}{{\partial \theta^{3} }}\frac{\partial w}{{\partial \theta }} + \frac{{\partial^{2} w}}{{\partial \theta^{2} }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }}} \right) - \frac{{A_{12} }}{2R}\left( {\frac{\partial w}{{\partial x}}} \right)^{2} - \frac{{A_{22} }}{{2R^{3} }}\left( {\frac{\partial w}{{\partial \theta }}} \right)^{2} + \frac{{A_{12} }}{R}w\frac{{\partial^{2} w}}{{\partial x^{2} }} - B_{11} \left( {\frac{{\partial^{2} w}}{{\partial x^{2} }}} \right)^{2} \hfill \\ - \frac{{2B_{12} }}{{R^{2} }}\frac{{\partial^{2} w}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }} - \frac{{4B_{66} }}{{R^{2} }}\left( {\frac{{\partial^{2} w}}{\partial x\partial \theta }} \right)^{2} + \frac{{A_{22} }}{{R^{3} }}w\frac{{\partial^{2} w}}{{\partial \theta^{2} }} - \frac{{B_{22} }}{{R^{4} }}\left( {\frac{{\partial^{2} w}}{{\partial \theta^{2} }}} \right)^{2} + A_{11} \frac{\partial w}{{\partial x}}\frac{{\partial w^{*} }}{\partial x}\frac{{\partial^{2} w}}{{\partial x^{2} }} \hfill \\ + \frac{{A_{12} }}{{R^{2} }}\frac{\partial w}{{\partial \theta }}\frac{{\partial w^{*} }}{\partial \theta }\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{2A_{66} }}{{R^{2} }}\left( {\frac{{\partial w^{2} }}{\partial x\partial \theta }\frac{{\partial w^{*} }}{\partial x}\frac{\partial w}{{\partial \theta }} + \frac{{\partial w^{2} }}{\partial x\partial \theta }\frac{\partial w}{{\partial x}}\frac{{\partial w^{*} }}{\partial \theta }} \right) + \frac{{A_{12} }}{{R^{2} }}\frac{\partial w}{{\partial x}}\frac{{\partial w^{*} }}{\partial x}\frac{{\partial^{2} w}}{{\partial \theta^{2} }} \hfill \\ \end{gathered} $$
(A12)
$$ \begin{gathered} P_{4} = \frac{1}{2}A_{11} \left( {\frac{\partial w}{{\partial x}}} \right)^{2} \frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{1}{{2R^{2} }}A_{12} \left( {\frac{\partial w}{{\partial \theta }}} \right)^{2} \frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{2A_{66} }}{{R^{2} }}\frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{\partial w}{{\partial x}}\frac{\partial w}{{\partial \theta }} \hfill \\ + \frac{1}{{2R^{2} }}A_{12} \left( {\frac{\partial w}{{\partial x}}} \right)^{2} \frac{{\partial^{2} w}}{{\partial \theta^{2} }} + \frac{1}{{2R^{4} }}A_{12} \left( {\frac{\partial w}{{\partial \theta }}} \right)^{2} \frac{{\partial^{2} w}}{{\partial \theta^{2} }} \hfill \\ \end{gathered} $$
(A13)
$$ Q_{3} = A_{11} \frac{\partial u}{{\partial x}}\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{2A_{66} }}{{R^{2} }}\frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{\partial u}{{\partial \theta }} + \frac{{A_{12} }}{{R^{2} }}\frac{{\partial^{2} w}}{{\partial \theta^{2} }}\frac{\partial u}{{\partial x}} $$
(A14)
$$ R_{3} = \frac{1}{R}\left( {A_{12} + \frac{{B_{12} }}{R}} \right)\frac{\partial v}{{\partial \theta }}\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{1}{R}\left( {2A_{66} + \frac{{4A_{66} }}{R}} \right)\frac{{\partial^{2} w}}{\partial x\partial \theta }\frac{\partial v}{{\partial x}} + \frac{1}{{R^{3} }}\left( {A_{22} + \frac{{B_{22} }}{R}} \right)\frac{{\partial^{2} w}}{{\partial \theta^{2} }}\frac{\partial v}{{\partial \theta }} $$
(A15)

Appendix 2

$$ l_{11} = - \frac{{{\mkern 1mu} \left( {A_{66} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + A_{11} {\mkern 1mu} \pi^{2} {\mkern 1mu} R^{2} {\mkern 1mu} m^{2} } \right)}}{{{\mkern 1mu} L^{2} {\mkern 1mu} R^{2} {\mkern 1mu} }} $$
(B1)
$$ l_{12} = \frac{{{\mkern 1mu} \pi mn\left( {B_{12} + 2{\mkern 1mu} B_{66} + A_{12} {\mkern 1mu} R + A_{66} {\mkern 1mu} R} \right)}}{{{\mkern 1mu} L{\mkern 1mu} R^{2} {\mkern 1mu} }} $$
(B2)
$$ l_{13} = \frac{{\pi {\mkern 1mu} m\left( {B_{12} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + 2{\mkern 1mu} B_{66} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + A_{12} {\mkern 1mu} L^{2} {\mkern 1mu} R + B_{11} {\mkern 1mu} R^{2} {\mkern 1mu} m^{2} {\mkern 1mu} \pi^{2} } \right)}}{{L^{3} {\mkern 1mu} R^{2} {\mkern 1mu} }} $$
(B3)
$$ l_{21} = \frac{{\pi mn\left( {B_{12} + 2{\mkern 1mu} B_{66} + A_{12} {\mkern 1mu} R + A_{66} {\mkern 1mu} R} \right)}}{{L{\mkern 1mu} R^{2} {\mkern 1mu} }} $$
(B4)
$$ l_{22} = - \frac{{\left( \begin{gathered} A_{22} {\mkern 1mu} L^{2} {\mkern 1mu} R^{2} {\mkern 1mu} n^{2} + 2{\mkern 1mu} B_{22} {\mkern 1mu} L^{2} {\mkern 1mu} R{\mkern 1mu} n^{2} + D_{22} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} \hfill \\ + A_{66} {\mkern 1mu} \pi^{2} {\mkern 1mu} R^{4} {\mkern 1mu} m^{2} + 4{\mkern 1mu} B_{66} {\mkern 1mu} \pi^{2} {\mkern 1mu} R^{3} {\mkern 1mu} m^{2} + 4{\mkern 1mu} D_{66} {\mkern 1mu} \pi^{2} {\mkern 1mu} R^{2} {\mkern 1mu} m^{2} \hfill \\ \end{gathered} \right)}}{{L^{2} {\mkern 1mu} R^{4} }} $$
(B5)
$$ l_{23} = - \frac{{n\left( \begin{gathered} D_{22} L^{2} n^{2} + B_{22} L^{2} R + A_{22} L^{2} R^{2} + B_{22} L^{2} Rn^{2} + B_{12} R^{3} m^{2} \pi^{2} \hfill \\ + 2B_{66} R^{3} m^{2} \pi^{2} + D_{12} R^{3} m^{2} \pi^{2} + 4D_{66} R^{2} m^{2} \pi^{2} \hfill \\ \end{gathered} \right)}}{{L^{2} R^{4} }} $$
(B6)
$$ l_{31} = \frac{{\pi {\mkern 1mu} m\left( {B_{12} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + 2{\mkern 1mu} B_{66} {\mkern 1mu} L^{2} {\mkern 1mu} n^{2} + A_{12} {\mkern 1mu} L^{2} {\mkern 1mu} R + B_{11} {\mkern 1mu} R^{2} {\mkern 1mu} m^{2} {\mkern 1mu} \pi^{2} } \right)}}{{L^{3} {\mkern 1mu} R^{2} {\mkern 1mu} }} $$
(B7)
$$ l_{32} = - \frac{{n\left( \begin{gathered} D_{22} L^{2} n^{2} + B_{22} L^{2} R + A_{22} L^{2} R^{2} + B_{22} L^{2} Rn^{2} + B_{12} R^{3} m^{2} \pi^{2} \hfill \\ + 2B_{66} R^{3} m^{2} \pi^{2} + D_{12} R^{3} m^{2} \pi^{2} + 4D_{66} R^{2} m^{2} \pi^{2} \hfill \\ \end{gathered} \right)}}{{L^{2} R^{4} }} $$
(B8)
$$ l_{33} = - \frac{\begin{gathered} D_{22} L^{4} n^{4} + A_{22} L^{4} R^{2} + 2B_{22} L^{4} Rn^{2} + D_{11} R^{4} m^{4} \pi^{4} + 2B_{12} L^{2} R^{3} m^{2} \pi^{2} \hfill \\ + L^{2} R^{4} m^{2} \pi^{2} ph - I_{0} L^{2} R^{4} V^{2} m^{2} \pi^{2} + 4D_{66} L^{2} R^{2} m^{2} n^{2} \pi^{2} \hfill \\ \end{gathered} }{{L^{4} R^{4} }} $$
(B9)
$$ p_{3} = - \frac{{3A_{22} W_{1} n^{4} }}{{16R^{4} }} - \frac{{3A_{11} W_{1} m^{4} \pi^{4} }}{{16L^{4} }} - \frac{{\pi A_{12} W_{1} m^{2} }}{{16L^{2} R}} - \frac{{3A_{12} W_{1} m^{2} n^{2} \pi^{2} }}{{16L^{2} R^{2} }} + \frac{{A_{66} W_{1} m^{2} n^{2} \pi^{2} }}{{4L^{2} R^{2} }} $$
(B10)
$$ p_{4} = \frac{{ - 96A_{22} L^{4} mn^{5} \pi^{2} - 96A_{11} R^{4} m^{5} n\pi^{6} - 192A_{12} L^{2} R^{2} m^{3} n^{3} \pi^{4} + 128A_{66} L^{2} R^{2} m^{3} n^{3} \pi^{4} }}{{1024L^{4} R^{4} mn\pi^{2} }} $$
(B11)
$$ p_{1} = p_{2} = n_{1} = n_{2} = 0 $$
(B12)

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Zhang, YW., She, GL. Nonlinear low-velocity impact response of graphene platelet-reinforced metal foam cylindrical shells under axial motion with geometrical imperfection. Nonlinear Dyn 111, 6317–6334 (2023). https://doi.org/10.1007/s11071-022-08186-9

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