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Comparative Study of Free Vibration for Carbon Nanotube-Reinforced Composite Plates Based on Various Higher-Order Plate Theories

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Abstract

A comparative investigation is made on the free vibration behavior of a carbon nanotube-reinforced composite (CNTRC) plate based on various higher-order shear deformation plate theories (HSDTs). For the CNTRC plates, uniformly distributed (Type UD) and functionally graded (Type Λ) reinforcements are considered where the material properties of CNTRC plates are assumed to be graded in the thickness direction and estimated through the rule of mixture. A set of unified nonlinear dynamics governing equations of the CNTRC rectangular plate considering different shear deformation function forms are established based on von Kármán-type geometric nonlinearity. Then by applying Galerkin discretization method, the nonlinear governing equations of a simply-supported CNTRC plate are simplified into a set of ordinary differential equations involving independent time variable. Finally by using the harmonic balance method, the influences of the parameters such as nanotube volume fraction, the plate width-to-thickness ratio and the plate aspect ratio on the natural frequencies and nonlinear free vibration behaviors of CNTRC plates with various higher-order plate theories are analyzed comparatively.

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ACKNOWLEDGMENTS

Authors gratefully acknowledge the support of the Scientific Research Fund of Hunan Provincial Education Department (no. 16A003).

Author information

Authors and Affiliations

  1. School of Civil Engineering and Architecture, Changsha University of Science & Technology, 410114, Changsha, China

    Deliang Chen & Xiongfeng Peng

  2. Zoomlion Heavy Industry Science&Technology Co., Ltd., 410114, Changsha, China

    Lu Zhou

Authors
  1. Deliang Chen
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  2. Lu Zhou
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  3. Xiongfeng Peng
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Corresponding authors

Correspondence to Deliang Chen, Lu Zhou or Xiongfeng Peng.

Appendices

APPENDIX A

$$\begin{gathered} {{L}_{{11}}}\left( {} \right) = {{A}_{{11}}}{{\left( {} \right)}_{{,xx}}} + {{A}_{{66}}}{{\left( {} \right)}_{{,yy}}}, \\ {{L}_{{12}}}\left( {} \right) = \left( {{{A}_{{12}}} + {{A}_{{66}}}} \right){{\left( {} \right)}_{{,xy}}}, \\ {{L}_{{13}}}\left( {} \right) = \left( {{{E}_{{11}}} - {{B}_{{11}}}} \right){{\left( {} \right)}_{{,xxx}}} + \left( {{{E}_{{12}}} + 2{{E}_{{66}}} - {{B}_{{12}}} - 2{{B}_{{66}}}} \right){{\left( {} \right)}_{{,xyy}}}, \\ {{J}_{{13}}}\left( {} \right) = {{A}_{{11}}}{{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,xx}}} + {{A}_{{66}}}{{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,yy}}} + \left( {{{A}_{{12}}} + {{A}_{{66}}}} \right){{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,xy}}}, \\ {{L}_{{14}}}\left( {} \right) = {{E}_{{11}}}{{\left( {} \right)}_{{,xx}}} + {{E}_{{66}}}{{\left( {} \right)}_{{,yy}}}, \\ {{L}_{{15}}}\left( {} \right) = \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right){{\left( {} \right)}_{{,xy}}}, \\ \end{gathered} $$
(A.1)
$$\begin{gathered} {{L}_{{22}}}\left( {} \right) = {{A}_{{66}}}{{\left( {} \right)}_{{,xx}}} + {{A}_{{22}}}{{\left( {} \right)}_{{,yy}}}, \\ {{L}_{{23}}}\left( {} \right) = \left( {{{E}_{{22}}} - {{B}_{{22}}}} \right){{\left( {} \right)}_{{,yyy}}} + \left( {{{E}_{{12}}} + 2{{E}_{{66}}} - {{B}_{{12}}} - 2{{B}_{{66}}}} \right){{\left( {} \right)}_{{,xxy}}}, \\ {{J}_{{23}}}\left( {} \right) = {{A}_{{22}}}{{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,yy}}} + {{A}_{{66}}}{{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,xx}}} + \left( {{{A}_{{12}}} + {{A}_{{66}}}} \right){{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,xy}}}, \\ {{L}_{{25}}}\left( {} \right) = {{E}_{{66}}}{{\left( {} \right)}_{{,xx}}} + {{E}_{{22}}}{{\left( {} \right)}_{{,yy}}}, \\ \end{gathered} $$
(A.2)
$$\begin{gathered} {{L}_{{33}}}\left( {} \right) = {{A}_{{55}}}{{\left( {} \right)}_{{,xx}}} + {{A}_{{44}}}{{\left( {} \right)}_{{,yy}}} + \left( {2{{F}_{{11}}} - {{D}_{{11}}} - {{H}_{{11}}}} \right){{\left( {} \right)}_{{,xxxx}}} + \left( {2{{F}_{{22}}} - {{D}_{{22}}} - {{H}_{{22}}}} \right){{\left( {} \right)}_{{,yyyy}}} \\ + \,\left( {4{{F}_{{12}}} + 8{{F}_{{66}}} - 2{{D}_{{12}}} - 4{{D}_{{66}}} - 2{{H}_{{12}}} - 4{{H}_{{66}}}} \right){{\left( {} \right)}_{{,xxyy}}}, \\ {{L}_{{34}}}\left( {} \right) = {{A}_{{55}}}{{\left( {} \right)}_{{,x}}} + \left( {{{F}_{{11}}} - {{H}_{{11}}}} \right){{\left( {} \right)}_{{,xxx}}} + \left( {{{F}_{{12}}} + 2{{F}_{{66}}} - {{H}_{{12}}} - 2{{H}_{{66}}}} \right){{\left( {} \right)}_{{,xyy}}}, \\ {{L}_{{35}}}\left( {} \right) = {{A}_{{44}}}{{\left( {} \right)}_{{,y}}} + \left( {{{F}_{{22}}} - {{H}_{{22}}}} \right){{\left( {} \right)}_{{,yyy}}} + \left( {{{F}_{{12}}} + 2{{F}_{{66}}} - {{H}_{{12}}} - 2{{H}_{{66}}}} \right){{\left( {} \right)}_{{,xxy}}}, \\ {{J}_{{31}}}\left( {} \right) = {{A}_{{11}}}{{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,xx}}} + {{A}_{{12}}}{{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,yy}}} + 2{{A}_{{66}}}{{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,xy}}} + {{A}_{{11}}}{{\left( {} \right)}_{{,xx}}}{{\left( {} \right)}_{{,x}}} \\ + \,{{A}_{{66}}}{{\left( {} \right)}_{{,yy}}}{{\left( {} \right)}_{{,x}}} + \left( {{{A}_{{12}}} + {{A}_{{66}}}} \right){{\left( {} \right)}_{{,xy}}}{{\left( {} \right)}_{{,y}}}, \\ {{J}_{{32}}}\left( {} \right) = {{A}_{{12}}}{{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,xx}}} + {{A}_{{22}}}{{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,yy}}} + 2{{A}_{{66}}}{{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,xy}}} + {{A}_{{66}}}{{\left( {} \right)}_{{,xx}}}{{\left( {} \right)}_{{,y}}} \\ + \,{{A}_{{22}}}{{\left( {} \right)}_{{,yy}}}{{\left( {} \right)}_{{,y}}} + \left( {{{A}_{{12}}} + {{A}_{{66}}}} \right){{\left( {} \right)}_{{,xy}}}{{\left( {} \right)}_{{,x}}}, \\ J_{{33}}^{1}\left( {} \right) = \left( {2{{B}_{{66}}} + 2{{E}_{{12}}} - 2{{B}_{{12}}} - 2{{E}_{{66}}}} \right){{\left( {} \right)}_{{,xx}}}{{\left( {} \right)}_{{,yy}}} + \left( {2{{B}_{{12}}} + 2{{E}_{{66}}} - 2{{E}_{{12}}} - 2{{B}_{{66}}}} \right)\left( {} \right)_{{,xy}}^{2}, \\ J_{{33}}^{2}\left( {} \right) = \frac{3}{2}{{A}_{{11}}}\left( {} \right)_{{,x}}^{2}{{\left( {} \right)}_{{,xx}}} + \frac{3}{2}{{A}_{{22}}}\left( {} \right)_{{,y}}^{2}{{\left( {} \right)}_{{,yy}}} + \left( {\frac{1}{2}{{A}_{{12}}} + {{A}_{{66}}}} \right)\left( {} \right)_{{,x}}^{2}{{\left( {} \right)}_{{,yy}}}, \\ + \left( {\frac{1}{2}{{A}_{{12}}} + {{A}_{{66}}}} \right)\left( {} \right)_{{,y}}^{2}{{\left( {} \right)}_{{,xx}}} + \left( {2{{A}_{{12}}} + 4{{A}_{{66}}}} \right){{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,xy}}}, \\ \end{gathered} $$
$$\begin{gathered} {{J}_{{34}}}\left( {} \right) = {{E}_{{11}}}{{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,xx}}} + {{E}_{{12}}}{{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,yy}}} + 2{{E}_{{66}}}{{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,xy}}} + {{E}_{{11}}}{{\left( {} \right)}_{{,xx}}}{{\left( {} \right)}_{{,x}}} \\ \, + {{E}_{{66}}}{{\left( {} \right)}_{{,yy}}}{{\left( {} \right)}_{{,x}}} + \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right){{\left( {} \right)}_{{,xy}}}{{\left( {} \right)}_{{,y}}}, \\ {{J}_{{35}}}\left( {} \right) = {{E}_{{12}}}{{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,xx}}} + {{E}_{{22}}}{{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,yy}}} + 2{{E}_{{66}}}{{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,xy}}} + {{E}_{{66}}}{{\left( {} \right)}_{{,xx}}}{{\left( {} \right)}_{{,y}}} \\ \, + {{E}_{{22}}}{{\left( {} \right)}_{{,yy}}}{{\left( {} \right)}_{{,y}}} + \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right){{\left( {} \right)}_{{,xy}}}{{\left( {} \right)}_{{,x}}}, \\ \end{gathered} $$
(A.3)
$$\begin{gathered} {{L}_{{43}}}\left( {} \right) = - {{L}_{{34}}}\left( {} \right), \\ {{J}_{{43}}}\left( {} \right) = {{E}_{{11}}}{{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,xx}}} + {{E}_{{66}}}{{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,yy}}} + \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right){{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,xy}}}, \\ {{L}_{{44}}}\left( {} \right) = - {{A}_{{55}}}\left( {} \right) + {{H}_{{11}}}{{\left( {} \right)}_{{,xx}}} + {{H}_{{66}}}{{\left( {} \right)}_{{,yy}}}, \\ {{L}_{{45}}}\left( {} \right) = \left( {{{H}_{{12}}} + {{H}_{{66}}}} \right){{\left( {} \right)}_{{,xy}}}, \\ \end{gathered} $$
(A.4)
$$\begin{gathered} {{L}_{{53}}}\left( {} \right) = - {{L}_{{35}}}\left( {} \right) \\ {{J}_{{53}}}\left( {} \right) = {{E}_{{22}}}{{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,yy}}} + {{E}_{{66}}}{{\left( {} \right)}_{{,y}}}{{\left( {} \right)}_{{,xx}}} + ,\left( {{{E}_{{12}}} + {{E}_{{66}}}} \right){{\left( {} \right)}_{{,x}}}{{\left( {} \right)}_{{,xy}}}, \\ {{L}_{{55}}}\left( {} \right) = - {{A}_{{44}}}\left( {} \right) + {{H}_{{66}}}{{\left( {} \right)}_{{,xx}}} + {{H}_{{22}}}{{\left( - \right)}_{{,yy}}}, \\ \end{gathered} $$
(A.5)

APPENDIX B

$$\begin{gathered} {{a}_{{11}}} = {{I}_{0}}\frac{{ab}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{a}_{{12}}} = {{I}_{3}}\frac{{ab}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{a}_{{13}}} = \left( {{{I}_{3}} - {{I}_{1}}} \right)\frac{{bm\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{a}_{{14}}} = {{A}_{{11}}}\frac{{b{{m}^{2}}{{\pi }^{2}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + {{A}_{{66}}}\frac{{a{{n}^{2}}{{\pi }^{2}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{a}_{{15}}} = \left( {{{A}_{{12}}} + {{A}_{{66}}}} \right)\frac{{mn{{\pi }^{2}}}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{a}_{{16}}} = \left( {{{E}_{{11}}} - {{B}_{{11}}}} \right)\frac{{b{{m}^{3}}{{\pi }^{3}}}}{{4{{a}^{2}}}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {{{E}_{{12}}} - {{B}_{{12}}} + 2{{E}_{{66}}} - 2{{B}_{{66}}}} \right)\frac{{m{{n}^{2}}{{\pi }^{3}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{a}_{{17}}} = {{A}_{{11}}}\frac{{k{{r}^{2}}{{\pi }^{3}}}}{{{{a}^{3}}}}p_{2}^{{rkm}}q_{6}^{{nls}} + {{A}_{{66}}}\frac{{k{{s}^{2}}{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{2}^{{rkm}}q_{6}^{{nls}} - \left( {{{A}_{{12}}} + {{A}_{{66}}}} \right)\frac{{kls{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{2}^{{rkm}}q_{2}^{{nls}}, \\ {{a}_{{18}}} = {{E}_{{11}}}\frac{{b{{m}^{2}}{{\pi }^{2}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + {{E}_{{66}}}\frac{{a{{n}^{2}}{{\pi }^{2}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{a}_{{19}}} = \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right)\frac{{mn{{\pi }^{2}}}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}. \\ {{b}_{{11}}} = {{I}_{0}}\frac{{ab}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{b}_{{12}}} = {{I}_{3}}\frac{{ab}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{b}_{{13}}} = \left( {{{I}_{3}} - {{I}_{1}}} \right)\frac{{an\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ \end{gathered} $$
(B.1)
$$\begin{gathered} {{b}_{{14}}} = \left( {{{A}_{{12}}} + {{A}_{{66}}}} \right)\frac{{mn{{\pi }^{2}}}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{b}_{{15}}} = {{A}_{{66}}}\frac{{b{{m}^{2}}{{\pi }^{2}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + {{A}_{{22}}}\frac{{a{{n}^{2}}{{\pi }^{2}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{b}_{{16}}} = \left( {{{E}_{{22}}} - {{B}_{{22}}}} \right)\frac{{a{{n}^{3}}{{\pi }^{3}}}}{{4{{b}^{2}}}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {{{E}_{{12}}} - {{B}_{{12}}} + 2{{E}_{{66}}} - 2{{B}_{{66}}}} \right)\frac{{{{m}^{2}}n{{\pi }^{3}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ \end{gathered} $$
$$\begin{gathered} {{b}_{{17}}} = {{A}_{{22}}}\frac{{l{{s}^{2}}{{\pi }^{3}}}}{{{{b}^{3}}}}p_{6}^{{mkr}}q_{2}^{{sln}} + {{A}_{{66}}}\frac{{l{{r}^{2}}{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{6}^{{mkr}}q_{2}^{{sln}} - \left( {{{A}_{{12}}} + {{A}_{{66}}}} \right)\frac{{klr{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{2}^{{mkr}}q_{2}^{{sln}}, \\ {{b}_{{18}}} = \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right)\frac{{mn{{\pi }^{2}}}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{b}_{{19}}} = {{E}_{{66}}}\frac{{b{{m}^{2}}{{\pi }^{2}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + {{E}_{{22}}}\frac{{a{{n}^{2}}{{\pi }^{2}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}. \\ \end{gathered} $$
(B.2)
$$\begin{gathered} {{c}_{{11}}} = - {{I}_{0}}\frac{{ab}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}} - \left( {{{I}_{2}} - 2{{I}_{4}} + {{I}_{5}}} \right)\left( {\frac{{b{{m}^{2}}{{\pi }^{2}}}}{{4a}} + \frac{{a{{n}^{2}}{{\pi }^{2}}}}{{4b}}} \right){{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{c}_{{12}}} = \left( {{{I}_{1}} - {{I}_{3}}} \right)\frac{{bm\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{c}_{{13}}} = \left( {{{I}_{1}} - {{I}_{3}}} \right)\frac{{an\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{c}_{{14}}} = \left( {{{I}_{4}} - {{I}_{5}}} \right)\frac{{bm\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{c}_{{15}}} = \left( {{{I}_{4}} - {{I}_{5}}} \right)\frac{{an\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{c}_{{16}}} = \left( {{{B}_{{11}}} - {{E}_{{11}}}} \right)\frac{{b{{m}^{3}}{{\pi }^{3}}}}{{4{{a}^{2}}}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {{{B}_{{12}}} + 2{{B}_{{66}}} - {{E}_{{12}}} - 2{{E}_{{66}}}} \right)\frac{{m{{n}^{2}}{{\pi }^{3}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{c}_{{17}}} = \left( {{{B}_{{22}}} - {{E}_{{22}}}} \right)\frac{{a{{n}^{3}}{{\pi }^{3}}}}{{4{{b}^{2}}}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {{{B}_{{12}}} + 2{{B}_{{66}}} - {{E}_{{12}}} - 2{{E}_{{66}}}} \right)\frac{{{{m}^{2}}n{{\pi }^{3}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{c}_{{18}}} = - {{A}_{{55}}}\frac{{b{{m}^{2}}{{\pi }^{2}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} - {{A}_{{44}}}\frac{{a{{n}^{2}}{{\pi }^{2}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {2{{F}_{{11}}} - {{D}_{{11}}} - {{H}_{{11}}}} \right)\frac{{b{{m}^{4}}{{\pi }^{4}}}}{{4{{a}^{3}}}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} \\ + \left( {2{{F}_{{12}}} + 4{{F}_{{66}}} - {{D}_{{12}}} - 2{{D}_{{66}}} - {{H}_{{12}}} - 2{{H}_{{66}}}} \right)\frac{{{{m}^{2}}{{n}^{2}}{{\pi }^{4}}}}{{2ab}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {2{{F}_{{22}}} - {{D}_{{22}}} - {{H}_{{22}}}} \right)\frac{{a{{n}^{4}}{{\pi }^{4}}}}{{4{{b}^{3}}}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ \end{gathered} $$
$$\begin{gathered} {{c}_{{19}}} = - {{A}_{{55}}}\frac{{bm\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {{{F}_{{11}}} - {{H}_{{11}}}} \right)\frac{{b{{m}^{3}}{{\pi }^{3}}}}{{4{{a}^{2}}}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {{{F}_{{12}}} + 2{{F}_{{66}}} - {{H}_{{12}}} - 2{{H}_{{66}}}} \right)\frac{{m{{n}^{2}}{{\pi }^{3}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{c}_{{20}}} = - {{A}_{{44}}}\frac{{an\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {{{F}_{{22}}} - {{H}_{{22}}}} \right)\frac{{a{{n}^{3}}{{\pi }^{3}}}}{{4{{b}^{2}}}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {{{F}_{{12}}} + 2{{F}_{{66}}} - {{H}_{{12}}} - 2{{H}_{{66}}}} \right)\frac{{{{m}^{2}}n{{\pi }^{3}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ \end{gathered} $$
$$\begin{gathered} c_{{21}}^{1} = \left( {2{{B}_{{66}}} + 2{{E}_{{12}}} - 2{{B}_{{12}}} - 2{{E}_{{66}}}} \right)\frac{{{{k}^{2}}{{s}^{2}}{{\pi }^{4}}}}{{{{a}^{2}}{{b}^{2}}}}p_{6}^{{mkr}}q_{6}^{{nls}}, \\ c_{{21}}^{2} = \left( {2{{B}_{{12}}} + 2{{E}_{{66}}} - 2{{E}_{{12}}} - 2{{B}_{{66}}}} \right)\frac{{{{k}^{2}}{{l}^{2}}{{\pi }^{4}}}}{{{{a}^{2}}{{b}^{2}}}}p_{2}^{{mk}}q_{2}^{{nl}}, \\ c_{{22}}^{1} = - {{A}_{{11}}}\frac{{3{{k}^{2}}{{r}^{2}}{{\pi }^{4}}}}{{2{{a}^{4}}}}p_{4}^{{mkr}}q_{8}^{{nls}} - {{A}_{{22}}}\frac{{3{{l}^{2}}{{s}^{2}}{{\pi }^{4}}}}{{2{{b}^{4}}}}p_{8}^{{mkr}}q_{4}^{{nls}} - \left( {\frac{1}{2}{{A}_{{12}}} + {{A}_{{66}}}} \right)\frac{{{{l}^{2}}{{r}^{2}}{{\pi }^{4}}}}{{{{a}^{2}}{{b}^{2}}}}p_{4}^{{mkr}}q_{8}^{{nls}} \\ - \left( {\frac{1}{2}{{A}_{{12}}} + {{A}_{{66}}}} \right)\frac{{{{k}^{2}}{{s}^{2}}{{\pi }^{4}}}}{{{{a}^{2}}{{b}^{2}}}}p_{8}^{{mkr}}q_{4}^{{nls}},\quad c_{{22}}^{2} = \left( {2{{A}_{{12}}} + 4{{A}_{{66}}}} \right)\frac{{ksij{{\pi }^{4}}}}{{{{a}^{2}}{{b}^{2}}}}p_{5}^{{mikr}}q_{5}^{{njls}}, \\ {{c}_{{23}}} = {{A}_{{11}}}\frac{{k{{r}^{2}}{{\pi }^{3}}}}{{{{a}^{3}}}}p_{6}^{{mkr}}q_{6}^{{nls}} + {{A}_{{12}}}\frac{{k{{s}^{2}}{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{6}^{{mkr}}q_{6}^{{nls}} + 2{{A}_{{66}}}\frac{{lrs{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{2}^{{mkr}}q_{2}^{{nls}} \\ - {{A}_{{11}}}\frac{{{{k}^{2}}r{{\pi }^{3}}}}{{{{a}^{3}}}}p_{2}^{{mkr}}q_{6}^{{nls}} - {{A}_{{66}}}\frac{{{{l}^{2}}r{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{2}^{{mkr}}q_{6}^{{nls}} - \left( {{{A}_{{12}}} + {{A}_{{66}}}} \right)\frac{{kls{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{6}^{{mkr}}q_{2}^{{nls}}, \\ \end{gathered} $$
$$\begin{gathered} {{c}_{{24}}} = {{A}_{{12}}}\frac{{l{{r}^{2}}{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{6}^{{mkr}}q_{6}^{{nls}} + {{A}_{{22}}}\frac{{l{{s}^{2}}{{\pi }^{3}}}}{{{{b}^{3}}}}p_{6}^{{mkr}}q_{6}^{{nls}} + 2{{A}_{{66}}}\frac{{krs{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{2}^{{mkr}}q_{2}^{{nls}} - {{A}_{{66}}}\frac{{{{k}^{2}}s{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{6}^{{mkr}}q_{2}^{{nls}} \\ \, - {{A}_{{22}}}\frac{{{{l}^{2}}s{{\pi }^{3}}}}{{{{b}^{3}}}}p_{6}^{{mkr}}q_{2}^{{nls}} - \left( {{{A}_{{12}}} + {{A}_{{66}}}} \right)\frac{{klr{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{2}^{{mkr}}q_{6}^{{nls}}, \\ \end{gathered} $$
$$\begin{gathered} {{c}_{{25}}} = {{E}_{{11}}}\frac{{k{{r}^{2}}{{\pi }^{3}}}}{{{{a}^{3}}}}p_{6}^{{mkr}}q_{6}^{{nls}} + {{E}_{{12}}}\frac{{k{{s}^{2}}{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{6}^{{mkr}}q_{6}^{{nls}} + 2{{E}_{{66}}}\frac{{lrs{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{2}^{{mkr}}q_{2}^{{nls}} \\ \, - {{E}_{{11}}}\frac{{{{k}^{2}}r{{\pi }^{3}}}}{{{{a}^{3}}}}p_{2}^{{mkr}}q_{6}^{{nls}} - {{E}_{{66}}}\frac{{{{l}^{2}}r{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{2}^{{mkr}}q_{6}^{{nls}} - \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right)\frac{{kls{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{6}^{{mkr}}q_{2}^{{nls}}, \\ {{c}_{{26}}} = {{E}_{{12}}}\frac{{l{{r}^{2}}{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{6}^{{mkr}}q_{6}^{{nls}} + {{E}_{{22}}}\frac{{l{{s}^{2}}{{\pi }^{3}}}}{{{{b}^{3}}}}p_{6}^{{mkr}}q_{6}^{{nls}} + 2{{E}_{{66}}}\frac{{krs{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{2}^{{mkr}}q_{2}^{{nls}} \\ \, - {{E}_{{66}}}\frac{{{{k}^{2}}s{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{6}^{{mkr}}q_{2}^{{nls}} - {{E}_{{22}}}\frac{{{{l}^{2}}s{{\pi }^{3}}}}{{{{b}^{3}}}}p_{6}^{{mkr}}q_{2}^{{nls}} - \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right)\frac{{klr{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{2}^{{mkr}}q_{6}^{{nls}}, \\ {{c}_{{27}}} = \frac{{qab}}{{mn{{\pi }^{2}}}}\left( {1 - \cos n\pi - \cos m\pi + \cos n\pi \cos m\pi } \right). \\ \end{gathered} $$
(B.3)
$$\begin{gathered} {{d}_{{11}}} = {{I}_{3}}\frac{{ab}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{d}_{{12}}} = {{I}_{5}}\frac{{ab}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{d}_{{13}}} = \left( {{{I}_{5}} - {{I}_{4}}} \right)\frac{{bm\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{d}_{{14}}} = {{E}_{{11}}}\frac{{b{{m}^{2}}{{\pi }^{2}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + {{E}_{{66}}}\frac{{a{{n}^{2}}{{\pi }^{2}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{d}_{{15}}} = \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right)\frac{{mn{{\pi }^{2}}}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{d}_{{16}}} = {{A}_{{55}}}\frac{{bm\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {{{H}_{{11}}} - {{F}_{{11}}}} \right)\frac{{b{{m}^{3}}{{\pi }^{3}}}}{{4{{a}^{2}}}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} \\ + \left( {{{H}_{{12}}} + 2{{H}_{{66}}} - {{F}_{{12}}} - 2{{F}_{{66}}}} \right)\frac{{m{{n}^{2}}{{\pi }^{3}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{d}_{{17}}} = {{E}_{{11}}}\frac{{k{{r}^{2}}{{\pi }^{3}}}}{{{{a}^{3}}}}p_{2}^{{rkm}}q_{6}^{{nls}} + {{E}_{{66}}}\frac{{k{{s}^{2}}{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{2}^{{rkm}}q_{6}^{{nls}} - \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right)\frac{{kls{{\pi }^{3}}}}{{a{{b}^{2}}}}p_{2}^{{rkm}}q_{2}^{{nls}}, \\ {{d}_{{18}}} = {{A}_{{55}}}\frac{{ab}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + {{H}_{{11}}}\frac{{b{{m}^{2}}{{\pi }^{2}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + {{H}_{{66}}}\frac{{a{{n}^{2}}{{\pi }^{2}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{d}_{{19}}} = \left( {{{H}_{{12}}} + {{H}_{{66}}}} \right)\frac{{mn{{\pi }^{2}}}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}. \\ {{e}_{{11}}} = {{I}_{3}}\frac{{ab}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{e}_{{12}}} = {{I}_{5}}\frac{{ab}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{e}_{{13}}} = \left( {{{I}_{5}} - {{I}_{4}}} \right)\frac{{an\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{e}_{{14}}} = \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right)\frac{{mn{{\pi }^{2}}}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}},\quad {{e}_{{15}}} = {{E}_{{66}}}\frac{{b{{m}^{2}}{{\pi }^{2}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + {{E}_{{22}}}\frac{{a{{n}^{2}}{{\pi }^{2}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{e}_{{16}}} = {{A}_{{44}}}\frac{{an\pi }}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + \left( {{{H}_{{22}}} - {{F}_{{22}}}} \right)\frac{{a{{n}^{3}}{{\pi }^{3}}}}{{4{{b}^{2}}}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} \\ \end{gathered} $$
(B.4)
$$\begin{gathered} + \left( {{{H}_{{12}}} + 2{{H}_{{66}}} - {{F}_{{12}}} - 2{{F}_{{66}}}} \right)\frac{{{{m}^{2}}n{{\pi }^{3}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ {{e}_{{17}}} = {{E}_{{22}}}\frac{{l{{s}^{2}}{{\pi }^{3}}}}{{{{b}^{3}}}}p_{6}^{{mkr}}q_{2}^{{sln}} + {{E}_{{66}}}\frac{{l{{r}^{2}}{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{6}^{{mkr}}q_{2}^{{sln}} - \left( {{{E}_{{12}}} + {{E}_{{66}}}} \right)\frac{{klr{{\pi }^{3}}}}{{{{a}^{2}}b}}p_{2}^{{mkr}}q_{2}^{{sln}}, \\ {{e}_{{18}}} = \left( {{{H}_{{12}}} + {{H}_{{66}}}} \right)\frac{{mn{{\pi }^{2}}}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}}, \\ \end{gathered} $$
$${{e}_{{19}}} = {{A}_{{44}}}\frac{{ab}}{4}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + {{H}_{{66}}}\frac{{b{{m}^{2}}{{\pi }^{2}}}}{{4a}}{{\delta }_{{mk}}}{{\delta }_{{nl}}} + {{H}_{{22}}}\frac{{a{{n}^{2}}{{\pi }^{2}}}}{{4b}}{{\delta }_{{mk}}}{{\delta }_{{nl}}},$$
(B.5)

where

$$\begin{gathered} p_{1}^{{mk}} = \int\limits_0^a {\cos \frac{{k\pi x}}{a}} \sin \frac{{m\pi x}}{a}{\text{dx;}}\quad q_{1}^{{nl}} = \int\limits_0^b {\cos \frac{{l\pi y}}{b}} \sin \frac{{n\pi y}}{b}{\text{dy;}} \\ p_{2}^{{rkm}} = \int\limits_0^a {\cos \frac{{k\pi x}}{a}} \sin \frac{{r\pi x}}{a}\cos \frac{{m\pi x}}{a}{\text{dx;}}\quad q_{2}^{{sln}} = \int\limits_0^b {\cos \frac{{l\pi y}}{b}} \sin \frac{{s\pi y}}{b}\cos \frac{{n\pi y}}{b}{\text{dy;}} \\ \end{gathered} $$
$$\begin{gathered} p_{2}^{{mk{\text{r}}}} = \int_0^a {\cos \frac{{k\pi x}}{a}\cos \frac{{r\pi x}}{a}} \sin \frac{{m\pi x}}{a}{\text{dx;}}\quad q_{2}^{{nls}} = \int\limits_0^b {\cos \frac{{l\pi y}}{b}\cos \frac{{s\pi y}}{b}} \sin \frac{{n\pi y}}{b}{\text{dy;}} \\ p_{3}^{{mk{\text{r}}}} = \int\limits_0^a {\cos \frac{{k\pi x}}{a}\sin \frac{{r\pi x}}{a}} \sin \frac{{m\pi x}}{a}{\text{dx;}}\quad q_{3}^{{nls}} = \int\limits_0^b {\cos \frac{{l\pi y}}{b}\sin \frac{{s\pi y}}{b}} \sin \frac{{n\pi y}}{b}{\text{dy;}} \\ p_{4}^{{mk{\text{r}}}} = \int\limits_0^a {\sin \frac{{k\pi x}}{a}{{{\cos }}^{2}}\frac{{r\pi x}}{a}} \sin \frac{{m\pi x}}{a}{\text{dx;}}\quad q_{4}^{{nls}} = \int\limits_0^b {\sin \frac{{l\pi y}}{b}} {{\cos }^{2}}\frac{{s\pi y}}{b}\sin \frac{{n\pi y}}{b}{\text{dy;}} \\ \end{gathered} $$
$$\begin{gathered} p_{5}^{{mik{\text{r}}}} = \int\limits_0^a {\cos \frac{{k\pi x}}{a}\sin \frac{{r\pi x}}{a}\cos \frac{{i\pi x}}{a}} \sin \frac{{m\pi x}}{a}{\text{dx;}} \\ q_{5}^{{njls}} = \int\limits_0^b {\sin \frac{{l\pi y}}{b}\cos \frac{{s\pi y}}{b}\cos \frac{{j\pi y}}{b}} \sin \frac{{n\pi x}}{b}{\text{dy;}} \\ \end{gathered} $$
(B.6)
$$\begin{gathered} p_{6}^{{mk{\text{r}}}} = \int\limits_0^a {\sin \frac{{k\pi x}}{a}\sin \frac{{r\pi x}}{a}} \sin \frac{{m\pi x}}{a}{\text{dx;}}\quad q_{6}^{{nls}} = \int\limits_0^b {\sin \frac{{l\pi y}}{b}\sin \frac{{s\pi y}}{b}} \sin \frac{{n\pi y}}{b}{\text{dy;}} \\ p_{7}^{{mk{\text{r}}}} = \int\limits_0^a {\sin \frac{{k\pi x}}{a}\cos \frac{{r\pi x}}{a}} \sin \frac{{m\pi x}}{a}{\text{dx;}}\quad q_{7}^{{nls}} = \int\limits_0^b {\sin \frac{{l\pi y}}{b}\cos \frac{{s\pi y}}{b}} \sin \frac{{n\pi y}}{b}{\text{dy;}} \\ p_{8}^{{mk{\text{r}}}} = \int\limits_0^a {\sin \frac{{k\pi x}}{a}} {{\sin }^{2}}\frac{{r\pi x}}{a}\sin \frac{{m\pi x}}{a}{\text{dx;}}\quad q_{8}^{{nls}} = \int\limits_0^b {\sin \frac{{l\pi y}}{b}{{{\sin }}^{2}}\frac{{s\pi y}}{b}} \sin \frac{{n\pi y}}{b}{\text{dy;}} \\ \end{gathered} $$

APPENDIX C

$${{k}_{0}} = - \frac{{{{c}_{{27}}}}}{{{{c}_{{11}}}}};$$
(C.1)
$$\begin{gathered} \omega _{0}^{2} = \frac{{{{c}_{{18}}}}}{{{{c}_{{11}}}}} + \frac{{{{c}_{{19}}}{{\Omega }_{{11}}}}}{{{{c}_{{11}}}{{\Omega }_{9}}}} + \frac{{{{c}_{{20}}}{{\Omega }_{5}}}}{{{{c}_{{11}}}{{\Omega }_{4}}}} - \frac{{{{c}_{{16}}}\Omega _{1}^{4}}}{{{{c}_{{11}}}\Omega _{1}^{1}}} - \frac{{{{c}_{{16}}}\Omega _{1}^{3}{{\Omega }_{5}}}}{{{{c}_{{11}}}\Omega _{1}^{1}{{\Omega }_{4}}}} - \frac{{{{c}_{{16}}}\Omega _{1}^{2}{{\Omega }_{{11}}}}}{{{{c}_{{11}}}\Omega _{1}^{1}{{\Omega }_{9}}}} + \frac{{{{c}_{{17}}}{{b}_{{14}}}\Omega _{1}^{4} - {{c}_{{17}}}{{b}_{{16}}}\Omega _{1}^{1}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}}} \\ {\kern 1pt} + \frac{{{{c}_{{17}}}{{b}_{{14}}}\Omega _{1}^{2}{{\Omega }_{{11}}} - {{c}_{{17}}}{{b}_{{18}}}\Omega _{1}^{1}{{\Omega }_{{11}}}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}{{\Omega }_{9}}}} + \frac{{{{c}_{{17}}}{{b}_{{14}}}\Omega _{1}^{3}{{\Omega }_{5}} - {{c}_{{17}}}{{b}_{{19}}}\Omega _{1}^{1}{{\Omega }_{5}}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}{{\Omega }_{4}}}}; \\ \end{gathered} $$
(C.2)
$$\begin{gathered} \chi = \frac{{{{c}_{{21}}}}}{{{{c}_{{11}}}}} + \frac{{{{c}_{{25}}}{{\Omega }_{{11}}}}}{{{{c}_{{11}}}{{\Omega }_{9}}}} + \frac{{{{c}_{{26}}}{{\Omega }_{5}}}}{{{{c}_{{11}}}{{\Omega }_{4}}}} + \frac{{{{c}_{{20}}}{{\Omega }_{6}}}}{{{{c}_{{11}}}{{\Omega }_{4}}}} + \frac{{{{c}_{{19}}}{{\Omega }_{{10}}}}}{{{{c}_{{11}}}{{\Omega }_{9}}}} - \frac{{{{c}_{{16}}}\Omega _{1}^{5}}}{{{{c}_{{11}}}\Omega _{1}^{1}}} - \frac{{{{c}_{{23}}}\Omega _{1}^{4}}}{{{{c}_{{11}}}\Omega _{1}^{1}}} - \frac{{{{c}_{{16}}}\Omega _{1}^{3}{{\Omega }_{6}}}}{{{{c}_{{11}}}\Omega _{1}^{1}{{\Omega }_{4}}}} - \frac{{{{c}_{{16}}}\Omega _{1}^{2}{{\Omega }_{{10}}}}}{{{{c}_{{11}}}\Omega _{1}^{1}{{\Omega }_{9}}}} \\ {\kern 1pt} - \frac{{{{c}_{{23}}}\Omega _{1}^{2}{{\Omega }_{{11}}}}}{{{{c}_{{11}}}\Omega _{1}^{1}{{\Omega }_{9}}}} - \frac{{{{c}_{{23}}}\Omega _{1}^{3}{{\Omega }_{5}}}}{{{{c}_{{11}}}\Omega _{1}^{1}{{\Omega }_{4}}}} + \frac{{{{c}_{{17}}}{{b}_{{14}}}\Omega _{1}^{5} - {{c}_{{17}}}{{b}_{{17}}}\Omega _{1}^{1}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}}} + \frac{{{{c}_{{24}}}{{b}_{{14}}}\Omega _{1}^{4} - {{c}_{{24}}}{{b}_{{16}}}\Omega _{1}^{1}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}}} \\ {\kern 1pt} + \frac{{{{c}_{{17}}}{{b}_{{14}}}\Omega _{1}^{2}{{\Omega }_{{10}}} - {{c}_{{17}}}{{b}_{{18}}}\Omega _{1}^{1}{{\Omega }_{{10}}}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}{{\Omega }_{9}}}} + \frac{{{{c}_{{17}}}{{b}_{{14}}}\Omega _{1}^{3}{{\Omega }_{6}} - {{c}_{{17}}}{{b}_{{19}}}\Omega _{1}^{1}{{\Omega }_{6}}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}{{\Omega }_{4}}}} \\ {\kern 1pt} + \frac{{{{c}_{{24}}}{{b}_{{14}}}\Omega _{1}^{2}{{\Omega }_{{11}}} - {{c}_{{24}}}{{b}_{{18}}}\Omega _{1}^{1}{{\Omega }_{{11}}}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}{{\Omega }_{9}}}} + \frac{{{{c}_{{24}}}{{b}_{{14}}}\Omega _{1}^{3}{{\Omega }_{5}} - {{c}_{{24}}}{{b}_{{19}}}\Omega _{1}^{1}{{\Omega }_{5}}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}{{\Omega }_{4}}}} \\ \end{gathered} $$
(C.3)
$$\begin{gathered} {\kern 1pt} \lambda = \frac{{{{c}_{{22}}}}}{{{{c}_{{11}}}}} + \frac{{{{c}_{{25}}}{{\Omega }_{{10}}}}}{{{{c}_{{11}}}{{\Omega }_{9}}}} + + \frac{{{{c}_{{26}}}{{\Omega }_{6}}}}{{{{c}_{{11}}}{{\Omega }_{4}}}} - \frac{{{{c}_{{23}}}\Omega _{1}^{5}}}{{{{c}_{{11}}}\Omega _{1}^{1}}} - \frac{{{{c}_{{23}}}\Omega _{1}^{2}{{\Omega }_{{10}}}}}{{{{c}_{{11}}}\Omega _{1}^{1}{{\Omega }_{9}}}} - \frac{{{{c}_{{23}}}\Omega _{1}^{3}{{\Omega }_{6}}}}{{{{c}_{{11}}}\Omega _{1}^{1}{{\Omega }_{4}}}} + \frac{{{{c}_{{24}}}{{b}_{{14}}}\Omega _{1}^{5} - {{c}_{{24}}}{{b}_{{17}}}\Omega _{1}^{1}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}}} \\ {\kern 1pt} + \frac{{{{c}_{{24}}}{{b}_{{14}}}\Omega _{1}^{2}{{\Omega }_{{10}}} - {{c}_{{24}}}{{b}_{{18}}}\Omega _{1}^{1}{{\Omega }_{{10}}}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}{{\Omega }_{9}}}} + \frac{{{{c}_{{24}}}{{b}_{{14}}}\Omega _{1}^{3}{{\Omega }_{6}} - {{c}_{{24}}}{{b}_{{19}}}\Omega _{1}^{1}{{\Omega }_{6}}}}{{{{c}_{{11}}}{{b}_{{15}}}\Omega _{1}^{1}{{\Omega }_{4}}}}, \\ \end{gathered} $$
(C.4)

where Ωi (i = 1~11), \(\Omega _{1}^{j}\)(j = 1~11) are given by:

$$\begin{gathered} \Omega _{1}^{1} = {{a}_{{15}}}{{b}_{{14}}} - {{b}_{{15}}}{{a}_{{14}}};\quad \Omega _{1}^{2} = {{a}_{{15}}}{{b}_{{18}}} - {{b}_{{15}}}{{a}_{{18}}};\quad \Omega _{1}^{3} = {{a}_{{15}}}{{b}_{{19}}} - {{b}_{{15}}}{{a}_{{19}}};\quad \Omega _{1}^{4} = {{a}_{{15}}}{{b}_{{16}}} - {{b}_{{15}}}{{a}_{{16}}}; \\ \Omega _{1}^{5} = {{a}_{{15}}}{{b}_{{17}}} - {{b}_{{15}}}{{a}_{{17}}};\quad \Omega _{1}^{6} = {{d}_{{15}}}{{b}_{{14}}} - {{b}_{{15}}}{{d}_{{14}}};\quad \Omega _{1}^{7} = {{e}_{{15}}}{{b}_{{14}}} - {{b}_{{15}}}{{e}_{{14}}};\quad \Omega _{1}^{8} = {{d}_{{18}}}{{b}_{{15}}} - {{b}_{{18}}}{{d}_{{15}}}; \\ \Omega _{1}^{9} = {{e}_{{18}}}{{b}_{{15}}} - {{b}_{{18}}}{{e}_{{15}}};\quad \Omega _{1}^{{10}} = {{e}_{{19}}}{{b}_{{15}}} - {{b}_{{19}}}{{e}_{{15}}};\quad \Omega _{1}^{{11}} = {{d}_{{19}}}{{b}_{{15}}} - {{b}_{{19}}}{{d}_{{15}}};\quad \Omega _{1}^{{12}} = {{d}_{{16}}}{{b}_{{15}}} - {{b}_{{16}}}{{d}_{{15}}}; \\ \Omega _{1}^{{13}} = {{b}_{{15}}}{{e}_{{16}}} - {{e}_{{15}}}{{b}_{{16}}};\quad \Omega _{1}^{{14}} = {{d}_{{17}}}{{b}_{{15}}} - {{b}_{{17}}}{{d}_{{15}}};\quad \Omega _{1}^{{15}} = {{e}_{{17}}}{{b}_{{15}}} - {{b}_{{17}}}{{e}_{{15}}}; \\ {{\Omega }_{2}} = \frac{{\Omega _{1}^{6}\Omega _{1}^{2} + \Omega _{1}^{8}\Omega _{1}^{1}}}{{{{b}_{{15}}}\Omega _{1}^{1}}};\quad {{\Omega }_{3}} = \frac{{\Omega _{1}^{7}\Omega _{1}^{2} + \Omega _{1}^{9}\Omega _{1}^{1}}}{{{{b}_{{15}}}\Omega _{1}^{1}}};\quad {{\Omega }_{4}} = {{\Omega }_{2}}{{\Omega }_{8}} - {{\Omega }_{3}}{{\Omega }_{7}}; \\ {{\Omega }_{5}} = \frac{{(\Omega _{1}^{6}\Omega _{1}^{4} + \Omega _{1}^{{12}}\Omega _{1}^{1}){{\Omega }_{3}} - (\Omega _{1}^{7}\Omega _{1}^{4} + \Omega _{1}^{{13}}\Omega _{1}^{1}){{\Omega }_{2}}}}{{{{b}_{{15}}}\Omega _{1}^{1}}}; \\ {{\Omega }_{6}} = \frac{{(\Omega _{1}^{6}\Omega _{1}^{5} + \Omega _{1}^{{14}}\Omega _{1}^{1}){{\Omega }_{3}} - (\Omega _{1}^{7}\Omega _{1}^{5} + \Omega _{1}^{{15}}\Omega _{1}^{1}){{\Omega }_{2}}}}{{{{b}_{{15}}}\Omega _{1}^{1}}}; \\ {{\Omega }_{7}} = \frac{{\Omega _{1}^{6}\Omega _{1}^{3} + \Omega _{1}^{{11}}\Omega _{1}^{1}}}{{{{b}_{{15}}}\Omega _{1}^{1}}};\quad {{\Omega }_{8}} = \frac{{\Omega _{1}^{7}\Omega _{1}^{3} + \Omega _{1}^{{10}}{{\Omega }_{1}}}}{{{{b}_{{15}}}{{\Omega }_{1}}}};\quad {{\Omega }_{9}} = {{\Omega }_{3}}{{\Omega }_{7}} - {{\Omega }_{2}}{{\Omega }_{8}}; \\ {{\Omega }_{{10}}} = \frac{{(\Omega _{1}^{6}\Omega _{1}^{5} + \Omega _{1}^{{14}}\Omega _{1}^{1}){{\Omega }_{8}} - (\Omega _{1}^{7}\Omega _{1}^{5} + \Omega _{1}^{{15}}\Omega _{1}^{1}){{\Omega }_{7}}}}{{{{b}_{{15}}}\Omega _{1}^{1}}}; \\ {{\Omega }_{{11}}} = \frac{{(\Omega _{1}^{6}\Omega _{1}^{4} + \Omega _{1}^{{12}}\Omega _{1}^{1}){{\Omega }_{8}} - (\Omega _{1}^{7}\Omega _{1}^{4} + \Omega _{1}^{{13}}\Omega _{1}^{1}){{\Omega }_{7}}}}{{{{b}_{{15}}}\Omega _{1}^{1}}}. \\ \end{gathered} $$
(C.5)

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Chen, D., Zhou, L. & Peng, X. Comparative Study of Free Vibration for Carbon Nanotube-Reinforced Composite Plates Based on Various Higher-Order Plate Theories. Mech. Solids 58, 1828–1847 (2023). https://doi.org/10.3103/S0025654422600684

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