An analytical solution for vibration analysis of sandwich plates reinforced with graphene nanoplatelets

Abstract

In this study, the free vibration of a composite sandwich plate reinforced with graphene nanoplatelets (GPLs) enclosed by piezoelectric layers is investigated using an analytical solution. In the framework of the first-order shear deformation plate theory, multilayer functionally graded graphene platelets-reinforced composite plate is assumed. Applying modified Halpin–Tsai model and rule of mixtures, the effective Young’s modulus, mass density and Poisson’s ratio of nanocomposites are predicted. In each individual layer, the weight fraction of GPL nanofillers illustrates a layer-wise variation along the thickness direction either uniformly or non-uniformly GPLs dispersed. Based on Maxwell’s equation, the electric potential in a piezoelectric layer is considered for open and closed circuit boundary conditions. Coupled governing equations of motion and boundary conditions are derived by using the Hamilton’s principle. Four auxiliary scalar functions are introduced to decouple the governing equations of motion and boundary conditions, which are solved analytically by employing Levy-type boundary conditions. The effects of GPLs weight fraction, GPLs distribution patterns, number of layers and aspect ratio are examined in detail. The results show that the best way to predict the most effective reinforcement is to distribute more GPLs with a larger surface area near the top and bottom surfaces of the plate. Besides, adding a small amount of GPLs as reinforcing nanofillers can significantly improve the stiffness of the plate.

Appendices

Appendix A

In Eq. (25), the kth layer reinforced by GPLs is placed between the points $z = z_{k}$ and $z = z_{k + 1}$ in the thickness direction. In Eq. (26), the stiffness coefficients can be introduced as below:

$$ \begin{gathered} \left( {A_{11} ,A_{12} } \right) = \mathop \sum \limits_{k = 1}^{{N_{L} }} \mathop \int \limits_{{z_{k} }}^{{z_{k + 1} }} \left( {Q_{11}^{\left( k \right)} ,Q_{12}^{\left( k \right)} } \right){\text{d}}z + 2\mathop \int \limits_{h}^{{h + h_{p} }} \left( {\overline{c}_{11} ,\overline{c}_{12} } \right){\text{d}}z + \eta_{1} \hfill \\ A_{66} = \mathop \sum \limits_{k = 1}^{{N_{L} }} \mathop \int \limits_{{z_{k} }}^{{z_{k + 1} }} Q_{66}^{\left( k \right)} {\text{ d}}z + \mathop \int \limits_{h}^{{h + h_{p} }} \left( {\overline{c}_{11} - \overline{c}_{12} } \right) {\text{d}}z \hfill \\ \left( {B_{11} ,B_{12} ,B_{66} } \right) = \mathop \sum \limits_{k = 1}^{{N_{L} }} \mathop \int \limits_{{z_{k} }}^{{z_{k + 1} }} \left( {Q_{11}^{(k)} , Q_{12}^{(k)} , Q_{66}^{(k)} } \right)z{\text{d}}z \hfill \\ \left( {D_{11} ,D_{12} } \right) = \mathop \sum \limits_{k = 1}^{{N_{L} }} \mathop \int \limits_{{z_{k} }}^{{z_{k + 1} }} \left( {Q_{11}^{(k)} , Q_{12}^{(k)} } \right)z^{2} {\text{d}}z + 2\mathop \int \limits_{h}^{{h + h_{p} }} \left( {\overline{c}_{11} ,\overline{c}_{12} } \right)z^{2} {\text{d}}z + \eta_{2} \hfill \\ D_{66} = \mathop \sum \limits_{k = 1}^{{N_{L} }} \mathop \int \limits_{{z_{k} }}^{{z_{k + 1} }} Q_{66}^{(k)} z^{2} {\text{d}}z + \mathop \int \limits_{h}^{{h + h_{p} }} \left( {\overline{c}_{11} - \overline{c}_{12} } \right)z^{2} {\text{d}}z \hfill \\ A_{55} = \mathop \sum \limits_{k = 1}^{{N_{L} }} \mathop \int \limits_{{z_{k} }}^{{z_{k + 1} }} Q_{66}^{(k)} {\text{d}}z + 2\mathop \int \limits_{h}^{{h + h_{p} }} \overline{c}_{55} {\text{d}}z. \hfill \\ \end{gathered} $$

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Appendix B

In Eqs. (15) and (18), the definitions of constant quantities are expressed as follows:

$$ \Gamma_{1} = 2h_{p} \left( {e_{15} + \overline{e}_{31} } \right),\quad \Gamma_{2} = 2e_{15} h_{p} ,\quad \Gamma_{3} = \frac{{16\overline{\Xi }_{33} }}{{h_{p} }}, $$

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for open circuit condition,

$$ \begin{gathered} \eta_{1} = \frac{{2\overline{e}_{31}^{2} h_{p} }}{{\overline{\Xi }_{33} }}, \eta_{2} = \frac{{\overline{e}_{31}^{2} h_{p} \left( {2h + h_{p} } \right)\left( {h + h_{p} } \right)}}{{\overline{\Xi }_{33} }}, \overline{\Gamma }_{1} = \frac{{\overline{e}_{31} e_{15} h_{p}^{2} \left( {h + h_{p} } \right)}}{{\overline{\Xi }_{33} }} , \hfill \\ \tilde{\Gamma }_{1} = - \frac{8}{3}\overline{e}_{31} (3h + h_{p} ), \tilde{\Gamma }_{2} = \frac{16}{3}e_{15} h_{p} \hfill \\ \overline{\Gamma }_{2} = - \frac{{\overline{e}_{31} \Xi_{11} h_{p}^{2} }}{{\overline{\Xi }_{33} }}\left( {h + h_{p} } \right), \tilde{\Gamma }_{2} = - \frac{{16h_{p} \Xi_{11} }}{3}, \hfill \\ \end{gathered} $$

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and for closed circuit condition

$$ \begin{gathered} \eta_{1} = 0,\quad \eta_{2} = 0, \quad \overline{\Gamma }_{1} = 0,\quad \tilde{\Gamma }_{1} = - \frac{4}{3}\overline{e}_{31} h_{p} ,\quad \tilde{\Gamma }_{2} = - \frac{4}{3}e_{15} h_{p} \hfill \\ \overline{\Gamma }_{2} = 0,\quad \tilde{\Gamma }_{2} = - \frac{{4h_{p} \Xi_{11} }}{3}. \hfill \\ \end{gathered} $$

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Appendix C

$$ \begin{gathered} X_{1} = \frac{{K^{2} A_{55} }}{{\tilde{\Gamma }_{2} }}, X_{2} = - \frac{{\overline{\Gamma }_{1} }}{{\tilde{\Gamma }_{2} }}, X_{3} = \frac{{K^{2} A_{55} }}{{\tilde{\Gamma }_{2} }}, X4 = \frac{{I_{0} \omega_{m}^{2} }}{{\tilde{\Gamma }_{2} }}, \hfill \\ X_{5} = - \frac{{ - \frac{{B_{11}^{2} }}{{A_{11} }} + D_{11} + \left( {\tilde{\Gamma }_{2} - \tilde{\Gamma }_{1} } \right)X_{2} + \overline{\Gamma }_{1} }}{{ - \frac{{B_{11} I_{0} \omega_{m}^{2} }}{{A_{11} }} + I_{1} \omega_{m}^{2} }} \hfill \\ X_{6} = - \frac{{ - \frac{{B_{11} I_{1} \omega_{m}^{2} }}{{A_{11} }} - K^{2} A_{55} + \left( {\tilde{\Gamma }_{2} - \tilde{\Gamma }_{1} } \right)X_{1} + I_{2} \omega_{m}^{2} }}{{ - \frac{{B_{11} I_{0} \omega_{m}^{2} }}{{A_{11} }} + I_{1} \omega_{m}^{2} }} \hfill \\ X_{7} = - \frac{{ - K^{2} A_{55} + \left( {\tilde{\Gamma }_{2} - \tilde{\Gamma }_{1} } \right)X_{3} }}{{ - \frac{{B_{11} I_{0} \omega_{m}^{2} }}{{A_{11} }} + I_{1} \omega_{m}^{2} }}, X_{8} = - \frac{{\left( {\tilde{\Gamma }_{2} - \tilde{\Gamma }_{1} } \right)X_{4} }}{{ - \frac{{B_{11} I_{0} \omega_{m}^{2} }}{{A_{11} }} + I_{1} \omega_{m}^{2} }},X_{9} = - \frac{{\Gamma_{1} + \tilde{\Gamma }_{3} X_{1} }}{{\Gamma_{3} }}, \hfill \\ X_{10} = - \frac{{\overline{\Gamma }_{2} + \tilde{\Gamma }_{3} X_{2} }}{{\Gamma_{3} }}, X_{11} = - \frac{{\Gamma_{2} + \tilde{\Gamma }_{3} X_{3} }}{{\Gamma_{3} }} \hfill \\ X_{12} = - \frac{{\tilde{\Gamma }_{3} X_{4} }}{{\Gamma_{3} }}, X_{13} = \frac{{A_{55} K^{2} - \tilde{\Gamma }_{2} X_{12} }}{{\tilde{\Gamma }_{2} X_{10} }}, X_{14} = \frac{{K^{2} A_{55} }}{{\tilde{\Gamma }_{2} X_{10} }} \hfill \\ X_{15} = - \frac{{\tilde{\Gamma }_{2} X_{9} + \overline{\Gamma }_{1} }}{{\tilde{\Gamma }_{2} X_{10} }}, X_{16} = - \frac{{X_{11} }}{{X_{10} }}, X_{17} = \frac{{I_{0} \omega_{m}^{2} }}{{\tilde{\Gamma }_{2} X_{10} }} \hfill \\ \overline{X}_{13} = \frac{{ - I_{0} \omega_{m}^{2} X_{7} - A_{11} X_{8} }}{{A_{11} X_{5} }}, \overline{X}_{14} = \frac{{ - I_{0} \omega_{m}^{2} X_{6} - I_{1} \omega_{m}^{2} }}{{A_{11} X_{5} }}, \overline{X}_{15} = \frac{{ - I_{0} \omega_{m}^{2} X_{5} - A_{11} X_{6} - B_{11} }}{{A_{11} X_{5} }}, \hfill \\ \overline{X}_{16} = - \frac{{X_{7} }}{{X_{5} }}, \overline{X}_{17} = - \frac{{I_{0} \omega_{m}^{2} X_{8} }}{{A_{11} X_{5} }} \hfill \\ X_{18} = \frac{{X_{13} - \overline{X}_{13} }}{{\overline{X}_{15} - X_{15} }}, X_{19} = \frac{{X_{14} - \overline{X}_{14} }}{{\overline{X}_{15} - X_{15} }}, X_{20} = \frac{{X_{16} - \overline{X}_{16} }}{{\overline{X}_{15} - X_{15} }}, X_{21} = \frac{{X_{17} - \overline{X}_{17} }}{{\overline{X}_{15} - X_{15} }} \hfill \\ \zeta_{1} = \frac{{X_{20} }}{{X_{19}^{2} - X_{14} - X_{15} X_{19} }} \hfill \\ \zeta_{2} = - X_{19} \zeta_{1} + \zeta_{7} \hfill \\ \zeta_{3} = - X_{19} \zeta_{7} + \zeta_{8} - X_{20} \hfill \\ \zeta_{4} = - X_{19} \zeta_{8} + \zeta_{9} - X_{18} \hfill \\ \zeta_{5} = - X_{19} \zeta_{9} - X_{21} \hfill \\ \zeta_{6} = \zeta_{1} \hfill \\ \zeta_{7} = - \frac{{ - X_{15} X_{20} + X_{19} X_{20} - X_{16} + X_{18} }}{{X_{19}^{2} - X_{14} - X_{15} X_{19} }} \hfill \\ \zeta_{8} = - \frac{{ - X_{15} X_{18} + X_{18} X_{19} - X_{13} + X_{21} }}{{X_{19}^{2} - X_{14} - X_{15} X_{19} }} \hfill \\ \zeta_{9} = - \frac{{ - X_{15} X_{21} + X_{19} X_{21} - X_{17} }}{{X_{19}^{2} - X_{14} - X_{15} X_{19} }} \hfill \\ \overline{\zeta }_{1} = A_{66} \overline{\zeta }_{4} \hfill \\ \overline{\zeta }_{2} = I_{0} \omega_{m}^{2} \overline{\zeta }_{4} + A_{66} \overline{\zeta }_{5} + B_{66} \hfill \\ \overline{\zeta }_{3} = I_{0} \omega_{m}^{2} \overline{\zeta }_{5} + I_{1} \omega_{m}^{2} \hfill \\ \overline{\zeta }_{4} = \frac{{\frac{{B_{66}^{2} }}{{A_{66} }} - D_{66} }}{{\left( { - \frac{{I_{0} B_{66} }}{{A_{66} }} + I_{1} } \right)\omega_{m}^{2} }} \hfill \\ \overline{\zeta }_{5} = \frac{{\frac{{I_{1} \omega_{m}^{2} B_{66} }}{{A_{66} }} - I_{2} \omega_{m}^{2} + K^{2} A_{55} }}{{\left( { - \frac{{I_{0} B_{66} }}{{A_{66} }} + I_{1} } \right)\omega_{m}^{2} }}. \hfill \\ \end{gathered} $$

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