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Nonlinear dynamic analysis of the graphene platelets reinforced porous plate with magneto-electro-elastic sheets subjected to impact load

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Abstract

This paper studies the nonlinear dynamic characteristics of the simply supported and clamped graphene platelets reinforced porous plates with magneto-electro-elastic (MEE) sheets (GPLRP-MEE) subjected to axial impact loads. Three kinds of impacts, i.e. sinusoidal, exponential, and rectangular loads, are considered in this research. Grounded on the refined third-order shear deformation theory and the von Kármán nonlinearity, the nonlinear governing equations are transformed into a group of ordinary differential equations with the aid of the Galerkin method. Then, the fourth-order Runge–Kutta approach is adopted to obtain the nonlinear behaviours of the GPLRP-MEE plate under the impact load. After validating with the published literature, some parametric experiments are conducted to investigate the effects of the pulse load configurations, the internal structure of the GPLRP core layer, the dimensions of MEE sheets, the external magnetic and electric potentials, and the Winkler–Pasternak foundation moduli on the dynamic behaviours of the structure. According to the numerical results, it is revealed that the rectangular and sinusoidal loads pose a greater threat to the structure than the exponential one.

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Acknowledgements

The work presented in this paper has been supported by an Australian Government Research Training Program Scholarship and Australian Research Council project DP240102559, IH210100048, IH200100010, and DP210101353.

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Authors and Affiliations

  1. Centre for Infrastructure Engineering and Safety (CIES), School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW, 2052, Australia

    Yuhang Tian, Qingya Li, Yuan Feng & Wei Gao

  2. School of Mechanical and Mechatronic Engineering, University of Technology Sydney, Sydney, Australia

    Zhen Luo

  3. School of Engineering, Swinburne University of Technology, Hawthorn, VIC, 3122, Australia

    Dong Ruan

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  1. Yuhang Tian
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Appendices

Appendix 1. The reduced material coefficients of GPLRP-MEE plate

$$ \tilde{Q}_{11} (z) = \left\{ \begin{gathered} \frac{{E_{c} (z)}}{{1 - \nu_{c} (z)^{2} }}\quad \quad \quad \; - h_{c} /2 \le z \le h_{c} /2 \hfill \\ Q_{11}^{f} (z) - \frac{{Q_{13}^{f} (z)^{2} }}{{Q_{33}^{f} (z)}}\; - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2\; \hfill \\ \end{gathered} \right. $$
(63)
$$ \tilde{Q}_{12} (z) = \left\{ \begin{gathered} \frac{{E_{c} (z)\nu_{c} (z)}}{{1 - \nu_{c} (z)^{2} }}\quad \quad \quad \quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ Q_{12}^{f} (z) - \frac{{Q_{13}^{f} (z)Q_{23}^{f} (z)}}{{Q_{33}^{f} (z)}}\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
(64)
$$ \tilde{Q}_{22} (z) = \left\{ \begin{gathered} \frac{{E_{c} (z)}}{{1 - \nu_{c} (z)^{2} }}\quad \quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ Q_{22}^{f} (z) - \frac{{Q_{23}^{f} (z)^{2} }}{{Q_{33}^{f} (z)}}\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
(65)
$$ \tilde{Q}_{44} (z) = \left\{ \begin{gathered} \frac{{E_{c} (z)}}{{2[1 + \nu_{c} (z)]}}\quad \; - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \tilde{Q}_{44} (z)\quad \quad \quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
(66)
$$ \tilde{Q}_{55} (z) = \left\{ \begin{gathered} \frac{{E_{c} (z)}}{{2[1 + \nu_{c} (z)]}}\quad \; - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \tilde{Q}_{55} (z)\quad \quad \quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
(67)
$$ \tilde{Q}_{66} (z) = \left\{ \begin{gathered} \frac{{E_{c} (z)}}{{2[1 + \nu_{c} (z)]}}\quad \; - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \tilde{Q}_{66} (z)\quad \quad \quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
(68)
$$ \tilde{e}_{31} (z) = \left\{ \begin{gathered} 0\quad \quad \quad \quad \quad \quad \quad \quad \;\, - h_{c} /2 \le z \le h_{c} /2 \hfill \\ e_{31}^{f} (z) - \frac{{Q_{13}^{f} (z)e_{33}^{f} (z)}}{{Q_{33}^{f} (z)}}\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
(69)
$$ \tilde{e}_{32} (z) = \left\{ \begin{gathered} 0\quad \quad \quad \quad \quad \quad \quad \quad \quad \; - h_{c} /2 \le z \le h_{c} /2\; \hfill \\ e_{32}^{f} (z) - \frac{{Q_{23}^{f} (z)e_{33}^{f} (z)}}{{Q_{33}^{f} (z)}}\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
(70)
$$ \tilde{e}_{24} (z) = \left\{ \begin{gathered} 0\quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ e_{24}^{f} (z)\quad \; - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{e}_{15} (z) = \left\{ \begin{gathered} 0\quad \quad \;\; - h_{c} /2 \le z \le h_{c} /2 \hfill \\ e_{15}^{f} (z)\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{q}_{31} (z) = \left\{ \begin{gathered} 0\quad \quad \quad \quad \quad \quad \quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ q_{31}^{f} (z) - \frac{{Q_{13}^{f} (z)q_{33}^{f} (z)}}{{Q_{33}^{f} (z)}}\quad \; - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{q}_{32} (z) = \left\{ \begin{gathered} 0\quad \quad \quad \quad \quad \quad \quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ q_{32}^{f} (z) - \frac{{Q_{23}^{f} (z)q_{33}^{f} (z)}}{{Q_{33}^{f} (z)}}\quad \; - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{q}_{24} (z) = \left\{ \begin{gathered} 0\quad \quad \quad \; - h_{c} /2 \le z \le h_{c} /2 \hfill \\ q_{24}^{f} (z)\quad \; - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{q}_{15} (z) = \left\{ \begin{gathered} 0\quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ q_{15}^{f} (z)\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{\chi }_{11} (z) = \left\{ \begin{gathered} 0\quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \chi_{11}^{f} (z)\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{\chi }_{22} (z) = \left\{ \begin{gathered} 0\quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \chi_{22}^{f} (z)\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{\chi }_{33} (z) = \left\{ \begin{gathered} 0\quad \quad \quad \quad \quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \chi_{33}^{f} (z) + \frac{{e_{33}^{f} (z)^{2} }}{{Q_{33}^{f} (z)}}\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{\lambda }_{11} (z) = \left\{ \begin{gathered} 0\quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \lambda_{11}^{f} (z)\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{\lambda }_{22} (z) = \left\{ \begin{gathered} 0\quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \lambda_{22}^{f} (z)\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{\lambda }_{33} (z) = \left\{ \begin{gathered} 0\quad \quad \quad \quad \quad \quad \quad \quad \; - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \lambda_{33}^{f} (z) + \frac{{q_{33}^{f} (z)e_{33}^{f} (z)}}{{Q_{33}^{f} (z)}}\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{\mu }_{11} (z) = \left\{ \begin{gathered} 0\quad \quad \quad - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \mu_{11}^{f} (z)\quad - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{\mu }_{22} (z) = \left\{ \begin{gathered} 0\quad \quad \quad \, - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \mu_{22}^{f} (z)\quad \, - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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$$ \tilde{\mu }_{33} (z) = \left\{ \begin{gathered} 0\quad \quad \quad \quad \quad \quad \quad \; - h_{c} /2 \le z \le h_{c} /2 \hfill \\ \mu_{33}^{f} (z) + \frac{{q_{33}^{f} (z)^{2} }}{{Q_{33}^{f} (z)}}\quad \, - h/2 \le z < - h_{c} /2, \, h_{c} /2 < z \le h/2 \hfill \\ \end{gathered} \right. $$
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Appendix 2. The coefficients in the expressions of the stress resultants

$$ \left( {A_{ij} ,B_{ij} ,D_{ij} ,E_{ij} ,F_{ij} ,G_{ij} ,H_{ij} } \right) = \int_{ - h/2}^{h/2} {Q_{ij} (1,z,z^{2} ,g,zg,1 - g^{\prime},g^{2} )dz} { (}i,j = 1,2,4,5,6) $$
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$$ \begin{array}{*{20}c} {A_{31}^{e} = \frac{\pi }{h}\int_{ - h/2}^{h/2} {\tilde{e}_{31} (z)\sin \left( {\frac{\pi z}{h}} \right)dz,} }{A_{32}^{e} = \frac{\pi }{h}\int_{ - h/2}^{h/2} {\tilde{e}_{32} (z)\sin \left( {\frac{\pi z}{h}} \right)dz,} } \\ {A_{31}^{eb} = \frac{2}{h}\varphi_{0} \int_{ - h/2}^{h/2} {\tilde{e}_{31} (z)dz,} }{A_{32}^{eb} = \frac{2}{h}\varphi_{0} \int_{ - h/2}^{h/2} {\tilde{e}_{32} (z)dz,} } \\ {A_{31}^{m} = \frac{\pi }{h}\int_{ - h/2}^{h/2} {\tilde{q}_{31} (z)\sin \left( {\frac{\pi z}{{h_{c} }}} \right)dz,} }{A_{32}^{m} = \frac{\pi }{h}\int_{ - h/2}^{h/2} {\tilde{q}_{32} (z)\sin \left( {\frac{\pi z}{h}} \right)dz,} } \\ {A_{31}^{mb} = \frac{2}{h}\psi_{0} \int_{ - h/2}^{h/2} {\tilde{q}_{31} (z)dz,} }{A_{32}^{mb} = \frac{2}{h}\psi_{0} \int_{ - h/2}^{h/2} {\tilde{q}_{32} (z)dz} } \\ \end{array} $$
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$$ \begin{array}{*{20}c} {B_{31}^{e} = \frac{\pi }{h}\int_{ - h/2}^{h/2} {ze_{31} (z)\sin \left( {\frac{\pi z}{{h_{c} }}} \right)dz,} }{B_{32}^{e} = \frac{\pi }{h}\int_{ - h/2}^{h/2} {ze_{32} (z)\sin \left( {\frac{\pi z}{h}} \right)dz,} } \\ {B_{31}^{eb} = \frac{2}{h}\varphi_{0} \int_{ - h/2}^{h/2} {ze_{31} (z)dz,} }{B_{32}^{eb} = \frac{2}{h}\varphi_{0} \int_{ - h/2}^{h/2} {ze_{32} (z)dz,} } \\ {B_{31}^{m} = \frac{\pi }{h}\int_{ - h/2}^{h/2} {zq_{31} (z)\sin \left( {\frac{\pi z}{{h_{c} }}} \right)dz,} }{B_{32}^{m} = \frac{\pi }{h}\int_{ - h/2}^{h/2} {zq_{32} (z)\sin \left( {\frac{\pi z}{h}} \right)dz,} } \\ {B_{31}^{mb} = \frac{2}{h}\psi_{0} \int_{ - h/2}^{h/2} {zq_{31} (z)dz,} }{B_{32}^{mb} = \frac{2}{h}\psi_{0} \int_{ - h/2}^{h/2} {zq_{32} (z)dz} } \\ \end{array} $$
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$$ \begin{array}{*{20}c} {E_{31}^{e} = \frac{\pi }{{h_{c} }}\int_{ - h/2}^{h/2} {g(z)\tilde{e}_{31} (z)\sin \left( {\frac{\pi z}{h}} \right)dz,} }{E_{32}^{e} = \frac{\pi }{h}\int_{ - h/2}^{h/2} {g(z)\tilde{e}_{32} (z)\sin \left( {\frac{\pi z}{h}} \right)dz,} } \\ {E_{31}^{eb} = \frac{2}{h}\varphi_{0} \int_{ - h/2}^{h/2} {g(z)\tilde{e}_{31} (z)dz,} }{E_{32}^{eb} = \frac{2}{h}\varphi_{0} \int_{ - h/2}^{h/2} {g(z)\tilde{e}_{32} (z)dz,} } \\ {E_{31}^{m} = \frac{\pi }{h}\int_{ - h/2}^{h/2} {g(z)\tilde{q}_{31} (z)\sin \left( {\frac{\pi z}{h}} \right)dz,} }{E_{32}^{m} = \frac{\pi }{h}\int_{ - h/2}^{h/2} {g(z)\tilde{q}_{32} (z)\sin \left( {\frac{\pi z}{h}} \right)dz,} } \\ {E_{31}^{mb} = \frac{2}{h}\psi_{0} \int_{ - h/2}^{h/2} {g(z)\tilde{q}_{31} (z)dz,} }{E_{32}^{mb} = \frac{2}{h}\psi_{0} \int_{ - h/2}^{h/2} {g(z)\tilde{q}_{32} (z)dz,} } \\ \end{array} $$
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$$ \begin{gathered} G_{24}^{e} = - \int_{ - h/2}^{h/2} {\left( {1 - g^{\prime } } \right)\tilde{e}_{24} \cos \left( {\frac{\pi z}{h}} \right)dz} \hfill \\ G_{24}^{m} = - \int_{ - h/2}^{h/2} {\left( {1 - g^{\prime } } \right)\tilde{q}_{24} \cos \left( {\frac{\pi z}{h}} \right)dz} \hfill \\ G_{15}^{e} = - \int_{ - h/2}^{h/2} {\left( {1 - g^{\prime } } \right)\tilde{e}_{15} \cos \left( {\frac{\pi z}{h}} \right)dz} \hfill \\ G_{15}^{m} = - \int_{ - h/2}^{h/2} {\left( {1 - g^{\prime } } \right)\tilde{q}_{15} \cos \left( {\frac{\pi z}{h}} \right)dz} \hfill \\ \end{gathered} $$
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Appendix 3. The coefficients in the expressions of the membrane strains

$$ \begin{gathered} \begin{array}{*{20}c} {A_{11}^{*} = \frac{{A_{22} }}{{A_{11} A_{22} - A_{12}^{2} }},} & {A_{12}^{*} = - \frac{{A_{12} }}{{A_{11} A_{22} - A_{12}^{2} }},} \\ {A_{13}^{*} = \frac{{A_{12} B_{12} - A_{22} B_{11} }}{{A_{11} A_{22} - A_{12}^{2} }},} & {A_{14}^{*} = \frac{{A_{12} B_{22} - A_{22} B_{12} }}{{A_{11} A_{22} - A_{12}^{2} }},} \\ {A_{15}^{*} = \frac{{A_{12} E_{12} - A_{22} E_{11} }}{{A_{11} A_{22} - A_{12}^{2} }},} & {A_{16}^{*} = \frac{{A_{12} E_{22} - A_{22} E_{12} }}{{A_{11} A_{22} - A_{12}^{2} }},} \\ {A_{17}^{*} = \frac{{A_{12} A_{32}^{e} - A_{22} A_{31}^{e} }}{{A_{11} A_{22} - A_{12}^{2} }},} & {A_{18}^{*} = \frac{{A_{12} A_{32}^{m} - A_{22} A_{31}^{m} }}{{A_{11} A_{22} - A_{12}^{2} }},} \\ \end{array} \hfill \\ A_{19}^{*} = \frac{{A_{12} A_{32}^{eb} - A_{22} A_{31}^{eb} + A_{12} A_{32}^{mb} - A_{22} A_{31}^{mb} }}{{A_{11} A_{22} - A_{12}^{2} }} \hfill \\ \end{gathered} $$
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$$ \begin{array}{*{20}l} {\begin{array}{*{20}l} {A_{21}^{*} = - \frac{{A_{12} }}{{A_{11} A_{22} - A_{12}^{2} }},} \hfill & {A_{22}^{*} = \frac{{A_{11} }}{{A_{11} A_{22} - A_{12}^{2} }},} \hfill \\ {A_{23}^{*} = \frac{{A_{12} B_{11} - A_{11} B_{12} }}{{A_{11} A_{22} - A_{12}^{2} }},} \hfill & {A_{24}^{*} = \frac{{A_{12} B_{12} - A_{11} B_{22} }}{{A_{11} A_{22} - A_{12}^{2} }},} \hfill \\ {A_{25}^{*} = \frac{{A_{12} E_{11} - A_{11} E_{12} }}{{A_{11} A_{22} - A_{12}^{2} }},} \hfill & {A_{26}^{*} = \frac{{A_{12} E_{12} - A_{11} E_{22} }}{{A_{11} A_{22} - A_{12}^{2} }},} \hfill \\ {A_{27}^{*} = \frac{{A_{12} A_{31}^{e} - A_{11} A_{32}^{e} }}{{A_{11} A_{22} - A_{12}^{2} }},} \hfill & {A_{28}^{*} = \frac{{A_{12} A_{31}^{m} - A_{11} A_{32}^{m} }}{{A_{11} A_{22} - A_{12}^{2} }},} \hfill \\ \end{array} } \hfill \\ {A_{29}^{*} = \frac{{A_{12} A_{31}^{eb} - A_{11} A_{32}^{eb} - A_{11} A_{32}^{mb} + A_{12} A_{31}^{mb} }}{{A_{11} A_{22} - A_{12}^{2} }}} \hfill \\ \end{array} $$
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$$ A_{31}^{*} = \frac{1}{{A_{66} }},A_{32}^{*} = - \frac{{B_{66} }}{{A_{66} }},A_{33}^{*} = - \frac{{E_{66} }}{{A_{66} }} $$
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Tian, Y., Li, Q., Feng, Y. et al. Nonlinear dynamic analysis of the graphene platelets reinforced porous plate with magneto-electro-elastic sheets subjected to impact load. Nonlinear Dyn 112, 1661–1690 (2024). https://doi.org/10.1007/s11071-023-09093-3

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