On the dynamics and wave propagation of reinforced composite nanosystem

Abstract

In this work, nonlocal dynamic formulation of a graphene nanoplatelets reinforced composite doubly curved micro/nano shell is presented based on Hamilton’s principle using a shear deformable model. The structure is composed of a honeycomb core integrated with graphene nanoplatelets reinforced face-sheets. The material properties of honeycomb core are computed using available formula in literature. Furthermore, material properties of composite reinforced face-sheets are assumed to vary along the thickness direction based on Halpin–Tsai micromechanical models and rule of mixture. The size-dependent governing equations of motion are derived through employing nonlocal equations. After verification of the formulation and solution procedure using a comparative study, the large parametric results are presented to discuss impact of main geometric, material and small scale parameters on the free vibration characteristics. As a main result of the present paper is this fact that the lowest frequencies are obtained for $\phi_{0} = {{h_{0} } \mathord{\left/ {\vphantom {{h_{0} } {l_{0} }}} \right. \kern-\nulldelimiterspace} {l_{0} }} = 0.6$. It is concluded that with increase of $\phi_{0}$ from small values, the mass is increased more than increase of stiffness that leads to a decrease in frequencies unlike higher values of $\phi_{0}$, in which an increase in structural stiffness is reached respect to a small increase in mass that leads to a main increase in frequencies.

Appendices

Appendix A

$$\begin{gathered} \left( {J_{1} ,J_{2} ,J_{3} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {Q_{{11b}} \left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {Q_{{11c}} \left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {Q_{{11t}} \left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z}} } \hfill \\ \left( {J_{4} ,J_{5} ,J_{6} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {Q_{{12b}} \left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {Q_{{12c}} \left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {Q_{{12t}} \left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z}} } \hfill \\ \left( {J_{8} ,J_{9} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {\frac{{k_{s} }}{{R_{{x_{1} }} }}Q_{{55b}} \left( {1,\frac{1}{{R_{{x_{1} }} }}} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {\frac{{k_{s} }}{{R_{{x_{1} }} }}Q_{{55c}} \left( {1,\frac{1}{{R_{{x_{1} }} }}} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {\frac{{k_{s} }}{{R_{{x_{1} }} }}Q_{{55t}} \left( {1,\frac{1}{{R_{{x_{1} }} }}} \right){\text{d}z}} } \hfill \\ \left( {J_{{11}} ,J_{{12}} ,J_{{13}} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {Q_{{66b}} \left( {1,z,\frac{1}{2}z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {Q_{{66c}} \left( {1,z,\frac{1}{2}z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)} \right){\text{d}z}} \hfill \\ + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {Q_{{66t}} \left( {1,z,\frac{1}{2}z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)} \right){\text{d}z}} \hfill \\ \left( {J_{{14}} ,J_{{15}} ,J_{{16}} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {\frac{1}{2}Q_{{66b}} z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)\left( {1,z,\frac{1}{2}z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)} \right){\text{d}z}} \hfill \\ + \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {\frac{1}{2}Q_{{66c}} z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)\left( {1,z,\frac{1}{2}z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)} \right){\text{d}z}} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {\frac{1}{2}Q_{{66t}} z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)\left( {1,z,\frac{1}{2}z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)} \right){\text{d}z}} \hfill \\ \left( {J_{{17}} ,J_{{18}} ,J_{{19}} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {Q_{{21b}} \left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {Q_{{21c}} \left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {Q_{{21t}} \left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z}} } \hfill \\ \left( {J_{{20}} ,J_{{21}} ,J_{{22}} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {Q_{{22b}} \left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {Q_{{22c}} \left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {Q_{{22t}} \left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z}} } \hfill \\ \left( {J_{{24}} ,J_{{25}} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {\frac{{k_{s} }}{{R_{{x_{2} }} }}Q_{{44b}} \left( {1,\frac{1}{{R_{{x_{2} }} }}} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {\frac{{k_{s} }}{{R_{{x_{2} }} }}Q_{{44c}} \left( {1,\frac{1}{{R_{{x_{2} }} }}} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {\frac{{k_{s} }}{{R_{{x_{2} }} }}Q_{{44t}} \left( {1,\frac{1}{{R_{{x_{2} }} }}} \right){\text{d}z}} } \hfill \\ \left( {J_{{27}} ,J_{{28}} ,J_{{29}} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {Q_{{11b}} z\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {Q_{{11c}} z\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {Q_{{11t}} z\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z}} } \hfill \\ \left( {J_{{30}} ,J_{{31}} ,J_{{32}} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {Q_{{12b}} z\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {Q_{{12c}} z\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {Q_{{12t}} z\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z}} } \hfill \\ \hfill \\ \end{gathered}$$

$$\left( {J_{34} ,J_{35} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {k_{s} Q_{55b} \left( {1,\frac{1}{{R_{{x_{1} }} }}} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {k_{s} Q_{55c} \left( {1,\frac{1}{{R_{{x_{1} }} }}} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {k_{s} Q_{55t} \left( {1,\frac{1}{{R_{{x_{1} }} }}} \right){\text{d}z}} }$$

$$\left( {J_{37} ,J_{38} ,J_{39} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {Q_{66b} z\left( {1,z,\frac{1}{2}z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {Q_{66c} z\left( {1,z,\frac{1}{2}z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)} \right){\text{d}z}}$$

$$\begin{aligned} & \qquad \qquad \qquad + \int\limits_{{\tfrac{{h_{c} }}{2}}}^{{\tfrac{{h_{c} }}{2} + h_{t} }} {Q_{{66t}} z\left( {1,z,\frac{1}{2}z\left( {\frac{1}{{R_{{x_{1} }} }} - \frac{1}{{R_{{x_{2} }} }}} \right)} \right){\text{d}z}} \\ & \left( {J_{{40}} ,J_{{41}} ,J_{{42}} } \right) = \int\limits_{{ - \tfrac{{h_{c} }}{2} - h_{b} }}^{{ - \tfrac{{h_{c} }}{2}}} {Q_{{21b}} z\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \tfrac{{h_{c} }}{2}}}^{{\tfrac{{h_{c} }}{2}}} {Q_{{21c}} z\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + \int\limits_{{\tfrac{{h_{c} }}{2}}}^{{\tfrac{{h_{c} }}{2} + h_{t} }} {Q_{{21t}} z\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z}} } \\ \end{aligned}$$

$$\left( {J_{43} ,J_{44} ,J_{45} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {Q_{22b} z\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {Q_{22c} z\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {Q_{22t} z\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z}} }$$

$$\left( {J_{47} ,J_{48} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {k_{s} Q_{44b} \left( {1,\frac{1}{{R_{{x_{2} }} }}} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {k_{s} Q_{44c} \left( {1,\frac{1}{{R_{{x_{2} }} }}} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {k_{s} Q_{44t} \left( {1,\frac{1}{{R_{{x_{2} }} }}} \right){\text{d}z}} }$$

$$\left( {J_{50} ,J_{51} ,J_{52} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {\frac{{Q_{11b} }}{{R_{{x_{1} }} }}\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {\frac{{Q_{11c} }}{{R_{{x_{1} }} }}\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {\frac{{Q_{11t} }}{{R_{{x_{1} }} }}\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z}} }$$

$$\left( {J_{53} ,J_{54} ,J_{55} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {\frac{{Q_{12b} }}{{R_{{x_{1} }} }}\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {\frac{{Q_{12c} }}{{R_{{x_{1} }} }}\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {\frac{{Q_{12t} }}{{R_{{x_{1} }} }}\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z}} }$$

$$\left( {J_{57} ,J_{58} ,J_{59} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {\frac{{Q_{21b} }}{{R_{{x_{2} }} }}\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {\frac{{Q_{21c} }}{{R_{{x_{2} }} }}\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {\frac{{Q_{21t} }}{{R_{{x_{2} }} }}\left( {1,\frac{1}{{R_{{x_{1} }} }},z} \right){\text{d}z}} }$$

$$\left( {J_{60} ,J_{61} ,J_{62} } \right) = \int\limits_{{ - \frac{{h_{c} }}{2} - h_{b} }}^{{ - \frac{{h_{c} }}{2}}} {\frac{{Q_{22b} }}{{R_{{x_{2} }} }}\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + } \int\limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} {\frac{{Q_{22c} }}{{R_{{x_{2} }} }}\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z} + \int\limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2} + h_{t} }} {\frac{{Q_{22t} }}{{R_{{x_{2} }} }}\left( {1,\frac{1}{{R_{{x_{2} }} }},z} \right){\text{d}z}} }$$

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Appendix B

$$K_{11} = \frac{{J_{1} m^{2} \pi^{2} }}{{a^{2} }} + \frac{{n^{2} \pi^{2} }}{{b^{2} }}\left( {J_{11} + J_{13} + J_{14} + J_{16} } \right) + J_{9}$$

$$K_{12} = \frac{{m\pi^{2} n}}{ab}\left( {J_{4} + J_{11} + J_{14} - J_{13} - J_{16} } \right)$$

$$K_{13} = - \frac{m\pi }{a}\left( {J_{2} + J_{5} + J_{8} } \right)$$

$$K_{14} = \frac{{J_{3} m^{2} \pi^{2} }}{{a^{2} }} + \frac{{n^{2} \pi^{2} }}{{b^{2} }}\left( {J_{12} + J_{15} } \right) - J_{8}$$

$$K_{15} = \frac{{m\pi^{2} n}}{ab}\left( {J_{6} + J_{12} + J_{15} } \right)$$

$$K_{21} = \frac{{m\pi^{2} n}}{ab}\left( {J_{11} + J_{13} + J_{17} - J_{14} - J_{16} } \right)$$

$$K_{22} = \frac{{m^{2} \pi^{2} }}{{a^{2} }}\left( {J_{11} + J_{16} - J_{13} - J_{14} } \right) + \frac{{J_{20} n^{2} \pi^{2} }}{{b^{2} }} + J_{25}$$

$$K_{23} = - \frac{n\pi }{b}\left( {J_{18} + J_{21} + J_{24} } \right)$$

$$K_{24} = \frac{{m\pi^{2} n}}{ab}\left( {J_{12} + J_{19} - J_{15} } \right)$$

$$K_{25} = \frac{{m^{2} \pi^{2} }}{{a^{2} }}\left( {J_{12} - J_{15} } \right) + \frac{{J_{22} n^{2} \pi^{2} }}{{b^{2} }} - J_{24}$$

$$K_{31} = - \frac{m\pi }{a}\left( {J_{35} + J_{50} + J_{57} } \right)$$

$$K_{32} = - \frac{n\pi }{b}\left( {J_{48} + J_{53} + J_{60} } \right)$$

$$K_{33} = \frac{{J_{34} m^{2} \pi^{2} }}{{a^{2} }} + \frac{{J_{47} n^{2} \pi^{2} }}{{b^{2} }} + J_{51} + J_{54} + J_{58} + J_{61} - k_{1} + k_{2} \left( { - \frac{{m^{2} \pi^{2} }}{{a^{2} }} - \frac{{n^{2} \pi^{2} }}{{b^{2} }}} \right)$$

$$K_{34} = - \frac{m\pi }{a}\left( {J_{52} + J_{59} - J_{34} } \right)$$

$$K_{35} = - \frac{n\pi }{b}\left( {J_{55} + J_{62} - J_{47} } \right)$$

$$K_{41} = \frac{{J_{27} m^{2} \pi^{2} }}{{a^{2} }} + \frac{{n^{2} \pi^{2} }}{{b^{2} }}\left( {J_{37} + J_{39} } \right) - J_{35}$$

$$K_{42} = \frac{{m\pi^{2} n}}{ab}\left( {J_{30} + J_{37} - J_{39} } \right)$$

$$K_{43} = - \frac{m\pi }{a}\left( {J_{28} + J_{31} - J_{34} } \right)$$

$$K_{44} = \frac{{J_{29} m^{2} \pi^{2} }}{{a^{2} }} + \frac{{J_{38} n^{2} \pi^{2} }}{{b^{2} }} + J_{34}$$

$$K_{45} = \frac{{m\pi^{2} n}}{ab}\left( {J_{32} + J_{38} } \right)$$

$$K_{51} = \frac{{m\pi^{2} n}}{ab}\left( {J_{37} + J_{39} + J_{40} } \right)$$

$$K_{52} = \frac{{m^{2} \pi^{2} }}{{a^{2} }}\left( {J_{37} - J_{39} } \right) + \frac{{J_{43} n^{2} \pi^{2} }}{{b^{2} }} - J_{48}$$

$$K_{53} = - \frac{n\pi }{b}\left( {J_{41} + J_{44} - J_{47} } \right)$$

$$K_{54} = \frac{{m\pi^{2} n}}{ab}\left( {J_{38} + J_{42} } \right)$$

$$K_{55} = \frac{{J_{38} m^{2} \pi^{2} }}{{a^{2} }} + \frac{{J_{45} n^{2} \pi^{2} }}{{b^{2} }} + J_{47}$$

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