Wave propagation behavior of coupled viscoelastic FG-CNTRPC micro plates subjected to electro-magnetic fields surrounded by orthotropic visco-Pasternak foundation

Abstract

Wave propagation in a coupled visco-elastic micro polyvinylidene fluoride (PVDF) plate reinforced by single-walled carbon nanotubes (SWCNTs) under an electro-magnetic field will be analyzed in this study. Quasi-3D sinusoidal shear deformation theory (SSDT) of plates is presumed in order to simulate plates. The system is surrounded by orthotropic visco-Pasternak foundation. Plates are assumed to have piezoelectric property and CNTs are applied to reinforce plates in different types so micro plates are functionally graded carbon nanotube-reinforced piezoelectric composite (FG-CNTRPC); therefore, electro-magnetic field can be used as a controlling parameter for system. At first, energy equation is written according to local piezoelasticity theory and strain energy, kinetic energy and work of external forces are calculated and motion equation of system will be calculated by applying Hamilton’s principle. Finally, motion equation will be solved by using analytical solution in order to obtaining velocity of wave propagation, cut-off and escape frequency of system in different types of wave propagation in system, such as in-phase, out-of-phase and one of plate fixed. After that, the effects of magnetic field, voltage, structure damping and distributions of CNTs on velocity of system and the effects of three type of wave propagation on cut-off and escape frequency of system will be studied. Results show that the effect of out-of-phase type on phase velocity and frequency of system is more than in-phase and one of plate fixed types.

Appendices

Appendix 1

$$\delta u: - \frac{{\partial N_{x}^{{}} }}{\partial x} - \frac{{\partial N_{xy}^{{}} }}{\partial y} + (1 - \mu^{2} \nabla^{2} )\left[ {m_{0} \frac{{\partial^{2} u}}{{\partial t^{2} }} - m_{1} \frac{{\partial^{3} w_{b} }}{{\partial x\partial t^{2} }} + (m_{3} - m_{1} )\frac{{\partial^{3} w_{s} }}{{\partial x\partial t^{2} }} - F_{mx} } \right],$$

(56)

$$\delta v: - \frac{{\partial N_{y}^{{}} }}{\partial y} - \frac{{\partial N_{xy}^{{}} }}{\partial x} + (1 - \mu^{2} \nabla^{2} )\left[ {m_{0} \frac{{\partial^{2} v}}{{\partial t^{2} }} - m_{1} \frac{{\partial^{3} w_{b} }}{{\partial y\partial t^{2} }} + (m_{3} - m_{1} )\frac{{\partial^{3} w_{s} }}{{\partial y\partial t^{2} }} - F_{my} } \right],$$

(57)

$$\begin{aligned} & \delta w_{b} : - \frac{{\partial^{2} M_{x}^{{}} }}{{\partial x^{2} }} - \frac{{\partial^{2} M_{y}^{{}} }}{{\partial y^{2} }} - 2 \frac{{\partial^{2} M_{xy}^{{}} }}{\partial y\partial x} + (1 - \mu^{2} \nabla^{2} )\left[ {m_{1} \frac{{\partial^{3} u}}{{\partial x\partial t^{2} }} + m_{1} \frac{{\partial^{3} v}}{{\partial y\partial t^{2} }} + m_{0} \frac{{\partial^{2} w_{b} }}{{\partial t^{2} }}} \right. \hfill \\ &\quad + m_{6} \frac{{\partial^{2} \varphi }}{{\partial t^{2} }} - m_{2} \frac{{\partial^{4} w_{b} }}{{\partial x^{2} \partial t^{2} }} - m_{2} \frac{{\partial^{4} w_{b} }}{{\partial y^{2} \partial t^{2} }} + (m_{5} - m_{2} )\frac{{\partial^{4} w_{s} }}{{\partial x^{2} \partial t^{2} }} - N_{xe} \frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - N_{ye} \frac{{\partial^{2} \varphi }}{{\partial y^{2} }} \hfill \\ &\quad + (m_{5} - m_{2} )\frac{{\partial^{4} w_{s} }}{{\partial y^{2} \partial t^{2} }} + m_{0} \frac{{\partial^{2} w_{s} }}{{\partial t^{2} }} - N_{ye} \frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - N_{xe} \frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - N_{ye} \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} - N_{xe} \frac{{\partial^{2} \varphi }}{{\partial x^{2} }} \hfill \\ &\quad \left. { + q - M_{mx,x} - M_{my,y} - F_{mz} } \right], \hfill \\ \end{aligned}$$

(58)

$$\begin{aligned} & \delta w_{s} : - \frac{{\partial^{2} M_{x}^{{}} }}{{\partial x^{2} }} - \frac{{\partial^{2} M_{y}^{{}} }}{{\partial y^{2} }} - 2 \frac{{\partial^{2} M_{xy}^{{}} }}{\partial y\partial x} + \frac{{\partial^{2} S_{x}^{{}} }}{{\partial x^{2} }} + \frac{{\partial^{2} S_{y}^{{}} }}{{\partial y^{2} }} + 2 \frac{{\partial^{2} S_{xy}^{{}} }}{\partial y\partial x} - \frac{{\partial Q_{xz}^{{}} }}{\partial x} - \frac{{\partial Q_{yz}^{{}} }}{\partial y} \hfill \\ &\quad + (1 - \mu^{2} \nabla^{2} )\left[ {m_{0} \frac{{\partial^{2} w_{b} }}{{\partial t^{2} }} + (m_{2} + m_{5} )} \right.\frac{{\partial^{4} w_{b} }}{{\partial x^{2} \partial t^{2} }} + (m_{4} - m_{2} + 2m_{5} )\frac{{\partial^{4} w_{s} }}{{\partial x^{2} \partial t^{2} }} \hfill \\ &\quad + (m_{1} + m_{3} )\frac{{\partial^{3} u}}{{\partial x\partial t^{2} }} + m_{6} \frac{{\partial^{2} \varphi }}{{\partial t^{2} }} + (m_{4} - m_{2} + 2m_{5} )\frac{{\partial^{4} w_{s} }}{{\partial y^{2} \partial t^{2} }} + (m_{1} + m_{3} )\frac{{\partial^{3} v}}{{\partial y\partial t^{2} }} \hfill \\ &\quad + m_{0} \frac{{\partial^{2} w_{s} }}{{\partial t^{2} }} - N_{xe} \frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - N_{ye} \frac{{\partial^{2} \varphi }}{{\partial y^{2} }} - N_{xe} \frac{{\partial^{2} \varphi }}{{\partial x^{2} }} + (m_{2} + m_{5} )\frac{{\partial^{4} w_{b} }}{{\partial y^{2} \partial t^{2} }} - N_{xe} \frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \hfill \\ &\quad \left. { - N_{ye} \frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - N_{ye} \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} + q - M_{mx,x} - M_{my,y} - F_{mz} } \right], \hfill \\ \end{aligned}$$

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$$\begin{aligned} & \delta \varphi : P_{z}^{{}} - \frac{{\partial Q_{xz}^{{}} }}{\partial x} - \frac{{\partial Q_{yz}^{{}} }}{\partial y} + (1 - \mu^{2} \nabla^{2} )\left[ {m_{7} \frac{{\partial^{2} \varphi }}{{\partial t^{2} }}} \right. - N_{xe} \frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} - N_{xe} \frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} - N_{xe} \frac{{\partial^{2} \varphi }}{{\partial x^{2} }} \hfill \\ &\quad + m_{6} \frac{{\partial^{2} w_{s} }}{{\partial t^{2} }} + m_{6} \frac{{\partial^{2} w_{b} }}{{\partial t^{2} }}\left. { - N_{ye} \frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} - N_{ye} \frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} - N_{ye} \frac{{\partial^{2} \varphi }}{{\partial y^{2} }} - F_{mz} } \right], \hfill \\ \end{aligned}$$

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$$\delta \phi^{{}} : \int_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} {\left( {g(z)\frac{{\partial D_{x}^{{}} }}{\partial x} + g(z)\frac{{\partial D_{y}^{{}} }}{\partial y} - g^{\prime}(z)D_{z}^{{}} } \right)} dz.$$

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Appendix 2

$$\begin{aligned} (U,V,W_{b} ,W_{s} ,\varPhi ) &= \frac{1}{h}(u,v,w_{b} ,w_{s} ,\varphi ), \ \varTheta = \frac{{\phi e_{31} }}{{A_{110} }},X = \frac{x}{a}, \ Y = \frac{y}{b},\ L_{x} = \frac{h}{a} ,\ e_{x} = \frac{\mu }{a}, \hfill \\ \lambda = \frac{a}{b},\tau = \frac{t}{h}\sqrt {\frac{{A_{110} }}{{I_{10} }}} ,\ (H_{X} ,H_{Y} ) = \frac{h \eta }{{A_{110} }}(H_{x}^{2} ,H_{y}^{2} ),\ (G_{F1} ,G_{F2} ) = \frac{1}{{A_{110} }}(G_{f\alpha } ,G_{f\beta } ), \hfill \\ C_{d} = c_{d} \sqrt {\frac{{h^{2} }}{{I_{10} A_{110} }}} , K_{W} = \frac{{K_{w} h^{2} }}{{A_{110} }},V_{0} = \frac{{v_{0} e_{31} }}{{A_{110} }} ,\ (\vartheta_{11} ,\vartheta_{22} ,\vartheta_{33} ) = \frac{{A_{110} }}{{he_{31}^{2} }}(\varepsilon_{11} ,\varepsilon_{22} ,\varepsilon_{33} ), K_{1} = k_{1} a, \hfill \\ K_{2} = k_{2} b, \omega = \varOmega h\sqrt {\frac{{I_{10} }}{{A_{110} }}} , \ V = v \sqrt {\frac{{I_{10} }}{{A_{110} }}} , (\zeta_{15} ,\zeta_{24} ,\zeta_{31} ,\zeta_{32} ,\zeta_{33} ) = \left( {\frac{{e_{15} }}{{e_{31} }},\frac{{e_{24} }}{{e_{31} }},\frac{{e_{31} }}{{e_{31} }},\frac{{e_{32} }}{{e_{31} }},\frac{{e_{33} }}{{e_{31} }}} \right), \hfill \\ \ \bar{g} = \frac{g}{h}\sqrt {\frac{{A_{110} }}{{I_{10} }}} ,\ (a_{ij} ,b_{ij} ,d_{ij} ,h_{ij} ,o_{ij} ,k_{ij} ,l_{ij} ) = \left( {\frac{{A_{ij} }}{{A_{110} }},\frac{{B_{ij} }}{{hA_{110} }},\frac{{D_{ij} }}{{h^{2} A_{110} }},\frac{{hH_{ij} }}{{A_{110} }},\frac{{O_{ij} }}{{A_{110} }},\frac{{K_{ij} }}{{A_{110} }},\frac{{h^{2} L_{ij} }}{{A_{110} }}} \right), \hfill \\ (\bar{M}_{0} ,\bar{M}_{1} ,\bar{M}_{2} ,\bar{M}_{3} ,\bar{M}_{4} ,\bar{M}_{5} ,\bar{M}_{6} ,\bar{M}_{7} ) = \left( {\frac{{m_{0} }}{{I_{10} }},\frac{{m_{1} }}{{I_{10} h}},\frac{{m_{2} }}{{I_{10} h^{2} }},\frac{{m_{3} }}{{I_{10} h}},\frac{{m_{4} }}{{I_{10} h^{2} }},\frac{{m_{5} }}{{I_{10} h^{2} }},\frac{{m_{6} }}{{I_{10} }},\frac{{m_{7} }}{{I_{10} }}} \right), \hfill \\ \end{aligned}$$

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Appendix 3

$$\begin{aligned} L_{11} &= \bar{g}a_{66} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} + a_{66} L_{x}^{2} \lambda^{2} K_{2}^{2} + \bar{g}a_{11} L_{x}^{2} \varOmega K_{1}^{2} + a_{11} L_{x}^{2} K_{1}^{2} + H_{Y} K_{1}^{2} L_{x}^{2} \hfill \\ &\quad + \overline{{M_{0} }} \varOmega^{2} + \lambda^{2} e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{2}^{2} + \lambda^{4} H_{Y} K_{2}^{4} L_{x}^{2} e_{x}^{2} + 2 \lambda^{2} H_{Y} K_{1}^{2} K_{2}^{2} L_{x}^{2} e_{x}^{2} \hfill \\ &\quad + \lambda^{2} H_{Y} K_{2}^{2} L_{x}^{2} + e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{1}^{2} + H_{Y} K_{1}^{4} L_{x}^{2} e_{x}^{2} , \hfill \\ \end{aligned}$$

(63)

$$L_{12} = - \bar{g}a_{66} L_{x}^{2} \lambda \varOmega K_{2} K_{1} - \bar{g}a_{12} L_{x}^{2} \lambda \varOmega K_{2} K_{1} - a_{66} L_{x}^{2} \lambda K_{2} K_{1} - a_{12} L_{x}^{2} \lambda K_{2} K_{1} ,$$

(64)

$$\begin{aligned} L_{13} &= - 2 ib_{66} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} - ib_{12} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} - i\bar{g}b_{11} L_{x}^{3} \varOmega K_{1}^{3} - iK_{1}^{3} L_{x}^{3} b_{11} \hfill \\ &\quad - 2 i\bar{g}b_{66} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} - i\overline{{M_{1} }} L_{x} \varOmega^{2} K_{1} - i\lambda^{2} e_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{2}^{2} K_{1} \hfill \\ &\quad - ie_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{1}^{3} - i\bar{g}b_{12} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} , \hfill \\ \end{aligned}$$

(65)

$$\begin{aligned} L_{14} &= - 2 ib_{66} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} - \frac{{ih_{11} L_{x}^{3} K_{1}^{3} }}{{\pi^{2} }} - \frac{{ih_{12} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} }}{{\pi^{2} }} + ie_{x}^{2} \overline{{M_{3} }} L_{x} \varOmega^{2} K_{1}^{3} \hfill \\ &\quad - ib_{11} L_{x}^{3} K_{1}^{3} + i\lambda^{2} e_{x}^{2} \overline{{M_{3} }} L_{x} \varOmega^{2} K_{2}^{2} K_{1} - i\bar{g}b_{11} L_{x}^{3} \varOmega K_{1}^{3} - \frac{{i\bar{g}h_{11} L_{x}^{3} \varOmega K_{1}^{3} }}{{\pi^{2} }} \hfill \\ &\quad - \frac{{2 i\bar{g}h_{66} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} }}{{\pi^{2} }} - 2 i\bar{g}b_{66} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} - \frac{{2 ih_{66} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} }}{{\pi^{2} }} \hfill \\ &\quad - i\bar{g}b_{12} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} - i\overline{{M_{1} }} L_{x} \varOmega^{2} K_{1} - ib_{12} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} + i\overline{{M_{3} }} L_{x} \varOmega^{2} K_{1} \hfill \\ &\quad - ie_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{1}^{3} - i\lambda^{2} e_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{2}^{2} K_{1} - \frac{{i\bar{g}h_{12} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} }}{{\pi^{2} }}, \hfill \\ \end{aligned}$$

(66)

$$L_{15} = - i\bar{g}h_{13} L_{x} \varOmega K_{1} - ih_{13} L_{x} K_{1} ,$$

(67)

$$L_{16} = 0 ,$$

(68)

$$L_{21} = - \bar{g}a_{66} L_{x}^{2} \lambda \varOmega K_{2} K_{1} - \bar{g}a_{12} L_{x}^{2} \lambda \varOmega K_{2} K_{1} - a_{66} L_{x}^{2} \lambda K_{2} K_{1} - a_{12} L_{x}^{2} \lambda K_{2} K_{1} ,$$

(69)

$$\begin{aligned} L_{22} &= \bar{g}a_{22} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} + a_{22} L_{x}^{2} \lambda^{2} K_{2}^{2} + \bar{g}a_{66} L_{x}^{2} \varOmega K_{1}^{2} + a_{66} L_{x}^{2} K_{1}^{2} + \overline{{M_{0} }} \varOmega^{2} \hfill \\ &\quad + \lambda^{2} e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{2}^{2} + \lambda^{4} H_{X} K_{2}^{4} L_{x}^{2} e_{x}^{2} + 2 \lambda^{2} H_{X} K_{1}^{2} K_{2}^{2} L_{x}^{2} e_{x}^{2} + H_{X} K_{1}^{4} L_{x}^{2} e_{x}^{2} \hfill \\ &\quad + H_{X} K_{1}^{2} L_{x}^{2} + e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{1}^{2} + \lambda^{2} H_{X} K_{2}^{2} L_{x}^{2} , \hfill \\ \end{aligned}$$

(70)

$$\begin{aligned} L_{23} &= 2 ib_{66} L_{x}^{3} \lambda K_{2} K_{1}^{2} + ib_{12} L_{x}^{3} \lambda K_{2} K_{1}^{2} + i\bar{g}b_{22} L_{x}^{3} \lambda^{3} \varOmega K_{2}^{3} + i\bar{g}b_{12} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} \hfill \\ &\quad + 2 i\bar{g}b_{66} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} + ie_{x}^{2} \overline{{M_{1} }} L_{x} \lambda \varOmega^{2} K_{2} K_{1}^{2} + i\lambda^{3} e_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{2}^{3} \hfill \\ &\quad + i\overline{{M_{1} }} L_{x} \lambda \varOmega^{2} K_{2} + ib_{22} L_{x}^{3} \lambda^{3} K_{2}^{3} , \hfill \\ \end{aligned}$$

(71)

$$\begin{aligned} L_{24} &= i\overline{{M_{1} }} L_{x} \lambda \varOmega^{2} K_{2} + \frac{{2 ih_{66} L_{x}^{3} \lambda K_{2} K_{1}^{2} }}{{\pi^{2} }} + ib_{22} L_{x}^{3} \lambda^{3} K_{2}^{3} + 2 i\bar{g}b_{66} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} \hfill \\ &\quad + \frac{{2 i\bar{g}h_{66} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} }}{{\pi^{2} }} + \frac{{i\bar{g}h_{12} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} }}{{\pi^{2} }} - ie_{x}^{2} \overline{{M_{3} }} L_{x} \lambda \varOmega^{2} K_{2} K_{1}^{2} \hfill \\ &\quad - i\overline{{M_{3} }} L_{x} \lambda \varOmega^{2} K_{2} i\bar{g}b_{12} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} + ie_{x}^{2} \overline{{M_{1} }} L_{x} \lambda \varOmega^{2} K_{2} K_{1}^{2} + \frac{{ih_{22} L_{x}^{3} \lambda^{3} K_{2}^{3} }}{{\pi^{2} }} \hfill \\ &\quad + i\bar{g}b_{22} L_{x}^{3} \lambda^{3} \varOmega K_{2}^{3} - i\lambda^{3} e_{x}^{2} \overline{{M_{3} }} L_{x} \varOmega^{2} K_{2}^{3} + \frac{{i\bar{g}h_{22} L_{x}^{3} \lambda^{3} \varOmega K_{2}^{3} }}{{\pi^{2} }} \hfill \\ &\quad + 2 ib_{66} L_{x}^{3} \lambda K_{2} K_{1}^{2} + i\lambda^{3} e_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{2}^{3} + \frac{{ih_{12} L_{x}^{3} \lambda K_{2} K_{1}^{2} }}{{\pi^{2} }} + ib_{12} L_{x}^{3} \lambda K_{2} K_{1}^{2} , \hfill \\ \end{aligned}$$

(72)

$$L_{25} = i\bar{g}h_{23} L_{x} \lambda \varOmega K_{2} + ih_{23} L_{x} \lambda K_{2} ,$$

(73)

$$L_{26} = 0 ,$$

(74)

$$\begin{aligned} L_{31} &= ib_{11} L_{x}^{3} K_{1}^{3} + i\bar{g}b_{12} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} + 2 i\bar{g}b_{66} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} + i\bar{g}b_{11} L_{x}^{3} \varOmega K_{1}^{3} \hfill \\ &\quad + 2 ib_{66} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} + ie_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{1}^{3} + i\lambda^{2} e_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{2}^{2} K_{1} \hfill \\ &\quad + ib_{12} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} + i\overline{{M_{1} }} L_{x} \varOmega^{2} K_{1} , \hfill \\ \end{aligned}$$

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$$\begin{aligned} L_{32} &= - i\bar{g}b_{22} L_{x}^{3} \lambda^{3} \varOmega K_{2}^{3} - 2 i\bar{g}b_{66} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} - i\bar{g}b_{12} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} \hfill \\ &\quad - i\overline{{M_{1} }} L_{x} \lambda \varOmega^{2} K_{2} - i\lambda^{3} e_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{2}^{3} - ie_{x}^{2} \overline{{M_{1} }} L_{x} \lambda \varOmega^{2} K_{2} K_{1}^{2} \hfill \\ &\quad - 2 ib_{66} L_{x}^{3} \lambda K_{2} K_{1}^{2} - ib_{12} L_{x}^{3} \lambda K_{2} K_{1}^{2} - ib_{22} L_{x}^{3} \lambda^{3} K_{2}^{3} , \hfill \\ \end{aligned}$$

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$$\begin{aligned} L_{33} &= 2 \bar{g}d_{12} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} + \frac{1}{12} e_{x}^{2} H_{Y} L_{x}^{4} K_{1}^{6} - H_{X} L_{x}^{2} \lambda^{2} K_{2}^{2} + H_{Y} L_{x}^{2} \lambda^{2} K_{2}^{2} \hfill \\ &\quad - e_{x}^{2} H_{Y} L_{x}^{2} K_{1}^{4} + e_{x}^{2} H_{X} L_{x}^{2} K_{1}^{4} + d_{22} L_{x}^{4} \lambda^{4} K_{2}^{4} + e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{1}^{2} \hfill \\ &\quad + \lambda^{4} e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{2}^{4} + 2 \lambda^{4} K_{2}^{4} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} + \frac{1}{6}\lambda^{4} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{4} K_{1}^{2} \hfill \\ &\quad + \frac{1}{6} \lambda^{2} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{2} K_{1}^{4} + \frac{1}{12}\lambda^{4} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{4} K_{1}^{2} + \frac{1}{12}e_{x}^{2} H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{4} \hfill \\ &\quad + \frac{1}{12} \lambda^{6} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{6} + \lambda^{4} e_{x}^{2} H_{Y} L_{x}^{2} K_{2}^{4} + \frac{1}{12} H_{Y} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} - \lambda^{4} e_{x}^{2} H_{X} L_{x}^{2} K_{2}^{4} \hfill \\ &\quad + \frac{1}{12} H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} + 2 \lambda^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} + 2 K_{1}^{4} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{1}^{4} \hfill \\ &\quad + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} + 4 \bar{g}d_{66} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} + 2 e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} K_{1}^{2} \hfill \\ &\quad + \lambda^{2} e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{2}^{2} + \bar{g}d_{11} L_{x}^{4} \varOmega K_{1}^{4} + 2 d_{12} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} + 4 d_{66} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} \hfill \\ &\quad + \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{1}^{2} + 2 K_{1}^{2} L_{x}^{2} V_{0} \zeta_{31} + \frac{1}{12} H_{X} L_{x}^{4} \lambda^{4} K_{2}^{4} + \frac{1}{12} H_{Y} L_{x}^{4} K_{1}^{4} - H_{Y} L_{x}^{2} K_{1}^{2} \hfill \\ &\quad + \bar{g}d_{22} L_{x}^{4} \lambda^{4} \varOmega K_{2}^{4} + \overline{{M_{2} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} + \overline{{M_{0} }} \varOmega^{2} + d_{11} L_{x}^{4} K_{1}^{4} + H_{X} L_{x}^{2} K_{1}^{2} \hfill \\ &\quad + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} - W_{elastic\ medium} , \hfill \\ \end{aligned}$$

(77)

$$\begin{aligned} L_{34} &= 4 \frac{{\bar{g}o_{66} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} + \frac{1}{6} \lambda^{2} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{2} K_{1}^{4} + \frac{1}{12} e_{x}^{2} H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{4} \hfill \\ &\quad + \frac{1}{12} \lambda^{4} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{4} K_{1}^{2} + \frac{1}{6} \lambda^{4} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{4} K_{1}^{2} + 2 \frac{{\lambda^{4} e_{x}^{2} L_{x}^{2} K_{2}^{4} H_{X} }}{\pi } \hfill \\ &\quad - 2 \frac{{H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{3} }} + d_{11} L_{x}^{4} K_{1}^{4} + H_{X} L_{x}^{2} K_{1}^{2} + 2 \frac{{\bar{g}o_{12} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} + \hfill \\ &\quad - 2 \frac{{\lambda^{6} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{6} }}{{\pi^{3} }} + \bar{g}d_{22} L_{x}^{4} \lambda^{4} \varOmega K_{2}^{4} + \frac{{\bar{g}o_{11} L_{x}^{4} \varOmega K_{1}^{4} }}{{\pi^{2} }} + 4 \frac{{o_{66} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} \hfill \\ &\quad - \lambda^{4} e_{x}^{2} \overline{{M_{5} }} L_{x}^{2} \varOmega^{2} K_{2}^{4} + \lambda^{4} e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{2}^{4} + 2 \lambda^{4} K_{2}^{4} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} \hfill \\ &\quad + 2 \frac{{\lambda^{2} e_{x}^{2} L_{x}^{2} K_{2}^{2} K_{1}^{2} H_{Y} }}{\pi } - 4 \frac{{\lambda^{2} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{2} K_{1}^{4} }}{{\pi^{3} }} + 2 \frac{{\lambda^{2} e_{x}^{2} L_{x}^{2} K_{2}^{2} K_{1}^{2} H_{X} }}{\pi } \hfill \\ &\quad - 2 \frac{{e_{x}^{2} H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{4} }}{{\pi^{3} }} - 2 \frac{{\lambda^{4} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{4} K_{1}^{2} }}{{\pi^{3} }} + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} \hfill \\ &\quad + 2 e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} K_{1}^{2} - 2 e_{x}^{2} \overline{{M_{5} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} K_{1}^{2} + \frac{{\bar{g}o_{22} L_{x}^{4} \lambda^{4} \varOmega K_{2}^{4} }}{{\pi^{2} }} \hfill \\ &\quad + 2 \bar{g}d_{12} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} + 2 \frac{{e_{x}^{2} L_{x}^{2} K_{1}^{4} H_{Y} }}{\pi } + 2 \frac{{L_{x}^{2} \lambda^{2} K_{2}^{2} H_{X} }}{\pi } - e_{x}^{2} L_{x}^{2} K_{1}^{4} H_{Y} \hfill \\ &\quad - 2 \frac{{e_{x}^{2} H_{Y} L_{x}^{4} K_{1}^{6} }}{{\pi^{3} }} - \lambda^{4} e_{x}^{2} H_{X} L_{x}^{2} K_{2}^{4} + \frac{1}{12}\lambda^{6} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{6} + \lambda^{4} e_{x}^{2} H_{Y} L_{x}^{2} K_{2}^{4} \hfill \\ &\quad + \frac{1}{12} H_{Y} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} + e_{x}^{2} H_{X} L_{x}^{2} K_{1}^{4} + \frac{1}{12} e_{x}^{2} H_{Y} L_{x}^{4} K_{1}^{6} + 2 \frac{{H_{Y} L_{x}^{2} K_{1}^{2} }}{\pi } \hfill \\ &\quad + H_{Y} L_{x}^{2} \lambda^{2} K_{2}^{2} + \frac{1}{12} H_{X} L_{x}^{4} \lambda^{4} K_{2}^{4} + 2 K_{1}^{2} L_{x}^{2} V_{0} \zeta_{31} + e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{1}^{2} \hfill \\ &\quad + d_{22} L_{x}^{4} \lambda^{4} K_{2}^{4} + 2 \lambda^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} + 2 K_{1}^{4} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{1}^{4} \hfill \\ &\quad - e_{x}^{2} \overline{{M_{5} }} L_{x}^{2} \varOmega^{2} K_{1}^{4} - \overline{{M_{5} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} + \lambda^{2} e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{2}^{2} + 2 d_{12} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} \hfill \\ &\quad + \frac{{o_{11} L_{x}^{4} K_{1}^{4} }}{{\pi^{2} }} - H_{Y} L_{x}^{2} K_{1}^{2} - 2 \frac{{H_{Y} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{3} }} + \frac{1}{12} H_{Y} L_{x}^{4} K_{1}^{4} - 2 \frac{{H_{X} L_{x}^{4} \lambda^{4} K_{2}^{4} }}{{\pi^{3} }} \hfill \\ &\quad - 4 \frac{{\lambda^{4} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{4} K_{1}^{2} }}{{\pi^{3} }} + 2 \frac{{o_{12} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} + 4 \bar{g}d_{66} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} \hfill \\ &\quad + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} - \overline{{M_{5} }} L_{x}^{2} \varOmega^{2} K_{1}^{2} + \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{1}^{2} - 2 \frac{{H_{Y} L_{x}^{4} K_{1}^{4} }}{{\pi^{3} }} \hfill \\ &\quad + \frac{1}{12} H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} + 4 d_{66} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} + \overline{{M_{2} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} - L_{x}^{2} \lambda^{2} K_{2}^{2} H_{X} \hfill \\ &\quad + \bar{g}d_{11} L_{x}^{4} \varOmega K_{1}^{4} + \frac{{o_{22} L_{x}^{4} \lambda^{4} K_{2}^{4} }}{{\pi^{2} }} + \overline{{M_{0} }} \varOmega^{2} - W_{elastic \ medium} , \hfill \\ \end{aligned}$$

(78)

$$\begin{aligned} L_{35} &= o_{23} L_{x}^{2} \lambda^{2} K_{2}^{2} + e_{x}^{2} \overline{{M_{6} }} \varOmega^{2} K_{1}^{2} + 2 K_{1}^{2} L_{x}^{2} V_{0} \zeta_{31} + 2 e_{x}^{2} \pi K_{1}^{2} H_{X} - 2 \frac{{L_{x}^{2} K_{1}^{2} H_{Y} }}{\pi } \hfill \\ &\quad + 2 e_{x}^{2} \pi K_{1}^{2} H_{Y} + \bar{g}o_{23} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} + 2 \lambda^{4} K_{2}^{4} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} - 2 \frac{{\lambda^{4} e_{x}^{2} L_{x}^{2} K_{2}^{4} H_{X} }}{\pi } \hfill \\ &\quad - \frac{{2L_{x}^{2} \lambda^{2} K_{2}^{2} H_{X} }}{\pi } + 2 \lambda^{2} e_{x}^{2} \pi K_{2}^{2} H_{Y} + \frac{{2L_{x}^{2} \lambda^{2} K_{2}^{2} H_{Y} }}{\pi } + \lambda^{2} e_{x}^{2} \overline{{M_{6} }} \varOmega^{2} K_{2}^{2} + \overline{{M_{6} }} \varOmega^{2} \hfill \\ &\quad + 2 \frac{{e_{x}^{2} L_{x}^{2} K_{1}^{4} H_{X} }}{\pi } + 2 \lambda^{2} e_{x}^{2} \pi K_{2}^{2} H_{X} - 2 \frac{{e_{x}^{2} L_{x}^{2} K_{1}^{4} H_{Y} }}{\pi } + 2 \lambda^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} \hfill \\ &\quad + \bar{g}o_{13} L_{x}^{2} \varOmega K_{1}^{2} + 2 \pi H_{Y} + 2 \pi H_{X} + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + o_{13} L_{x}^{2} K_{1}^{2} \hfill \\ &\quad + 2 \frac{{\lambda^{4} e_{x}^{2} L_{x}^{2} K_{2}^{4} H_{Y} }}{\pi } + 2 \frac{{L_{x}^{2} K_{1}^{2} H_{X} }}{\pi } + 2 K_{1}^{4} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} , \hfill \\ \end{aligned}$$

(79)

$$L_{36} = 2 \frac{{\zeta_{31} L_{x}^{2} K_{1}^{2} }}{\pi } + 2 \frac{{\zeta_{32} L_{x}^{2} \lambda^{2} K_{2}^{2} }}{\pi } ,$$

(80)

$$\begin{aligned} L_{41} &= - i\lambda^{2} e_{x}^{2} \overline{{M_{3} }} L_{x} \varOmega^{2} K_{2}^{2} K_{1} + 2 i\bar{g}b_{66} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} + i\lambda^{2} e_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{2}^{2} K_{1} \hfill \\ &\quad + ie_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{1}^{3} + \frac{{2 i\bar{g}h_{66} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} }}{{\pi^{2} }} + ib_{11} L_{x}^{3} K_{1}^{3} + \frac{{ih_{12} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} }}{{\pi^{2} }} \hfill \\ &\quad - i\lambda^{2} e_{x}^{2} \overline{{M_{3} }} L_{x} \varOmega^{2} K_{2}^{2} K_{1} + 2 i\bar{g}b_{66} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} + i\lambda^{2} e_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{2}^{2} K_{1} \hfill \\ &\quad + ie_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{1}^{3} + \frac{{2 i\bar{g}h_{66} L_{x}^{3} \lambda^{2} \varOmega K_{2}^{2} K_{1} }}{{\pi^{2} }} + ib_{11} L_{x}^{3} K_{1}^{3} + \frac{{ih_{12} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} }}{{\pi^{2} }} \hfill \\ &\quad + i\overline{{M_{1} }} L_{x} \varOmega^{2} K_{1} + i\overline{{M_{1} }} L_{x} \varOmega^{2} K_{1} - i\overline{{M_{3} }} L_{x} \varOmega^{2} K_{1} + ib_{12} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} \hfill \\ &\quad + ib_{12} L_{x}^{3} \lambda^{2} K_{2}^{2} K_{1} - i\overline{{M_{3} }} L_{x} \varOmega^{2} K_{1} , \hfill \\ \end{aligned}$$

(81)

$$\begin{aligned} L_{42} &= - i\lambda^{3} e_{x}^{2} \overline{{M_{1} }} L_{x} \varOmega^{2} K_{2}^{3} + ie_{x}^{2} \overline{{M_{3} }} L_{x} \lambda \varOmega^{2} K_{2} K_{1}^{2} - \frac{{i\bar{g}h_{12} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} }}{{\pi^{2} }} \hfill \\ &\quad - ib_{12} L_{x}^{3} \lambda K_{2} K_{1}^{2} + i\lambda^{3} e_{x}^{2} \overline{{M_{3} }} L_{x} \varOmega^{2} K_{2}^{3} - \frac{{ih_{12} L_{x}^{3} \lambda K_{2} K_{1}^{2} }}{{\pi^{2} }} + i\overline{{M_{3} }} L_{x} \lambda \varOmega^{2} K_{2} \hfill \\ &\quad - \frac{{2 ih_{66} L_{x}^{3} \lambda K_{2} K_{1}^{2} }}{{\pi^{2} }} - i\overline{{M_{1} }} L_{x} \lambda \varOmega^{2} K_{2} - ib_{22} L_{x}^{3} \lambda^{3} K_{2}^{3} - \frac{{ih_{22} L_{x}^{3} \lambda^{3} K_{2}^{3} }}{{\pi^{2} }} \hfill \\ &\quad - \frac{{2 i\bar{g}h_{66} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} }}{{\pi^{2} }} - ie_{x}^{2} \overline{{M_{1} }} L_{x} \lambda \varOmega^{2} K_{2} K_{1}^{2} - 2 i\bar{g}b_{66} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} \hfill \\ &\quad - i\bar{g}b_{12} L_{x}^{3} \lambda \varOmega K_{2} K_{1}^{2} - 2 ib_{66} L_{x}^{3} \lambda K_{2} K_{1}^{2} - \frac{{i\bar{g}h_{22} L_{x}^{3} \lambda^{3} \varOmega K_{2}^{3} }}{{\pi^{2} }} - i\bar{g}b_{22} L_{x}^{3} \lambda^{3} \varOmega K_{2}^{3} , \hfill \\ \end{aligned}$$

(82)

$$\begin{aligned} L_{43} &= \frac{1}{12}H_{Y} L_{x}^{4} K_{1}^{4} + \frac{1}{12} e_{x}^{2} H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{4} + d_{22} L_{x}^{4} \lambda^{4} K_{2}^{4} + \frac{{o_{11} L_{x}^{4} K_{1}^{4} }}{{\pi^{2} }} \hfill \\ &\quad + 2 K_{1}^{2} L_{x}^{2} V_{0} \zeta_{31} + \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{1}^{2} + e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{1}^{2} + \frac{1}{12} e_{x}^{2} H_{Y} L_{x}^{4} K_{1}^{6} \hfill \\ &\quad - H_{X} L_{x}^{2} \lambda^{2} K_{2}^{2} + H_{Y} L_{x}^{2} \lambda^{2} K_{2}^{2} + \frac{1}{12}H_{X} L_{x}^{4} \lambda^{4} K_{2}^{4} - e_{x}^{2} H_{Y} L_{x}^{2} K_{1}^{4} \hfill \\ &\quad + \frac{{\bar{g}o_{11} L_{x}^{4} \varOmega K_{1}^{4} }}{{\pi^{2} }} + 4 \frac{{o_{66} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} - \overline{{M_{5} }} L_{x}^{2} \varOmega^{2} K_{1}^{2} + e_{x}^{2} H_{X} L_{x}^{2} K_{1}^{4} \hfill \\ &\quad + \bar{g}d_{22} L_{x}^{4} \lambda^{4} \varOmega K_{2}^{4} + \frac{{2o_{12} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} + \lambda^{4} e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{2}^{4} + e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{1}^{4} \hfill \\ &\quad + 2 \lambda^{4} K_{2}^{4} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} + \frac{1}{6} \lambda^{4} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{4} K_{1}^{2} + \frac{1}{6}\lambda^{2} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{2} K_{1}^{4} \hfill \\ &\quad + \frac{1}{12} \lambda^{4} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{4} K_{1}^{2} + \overline{{M_{2} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} - e_{x}^{2} \overline{{M_{5} }} L_{x}^{2} \varOmega^{2} K_{1}^{4} + \overline{{M_{0} }} \varOmega^{2} \hfill \\ &\quad + 2 \frac{{\bar{g}o_{12} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} + 4 \frac{{\bar{g}o_{66} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} + \frac{{o_{22} L_{x}^{4} \lambda^{4} K_{2}^{4} }}{{\pi^{2} }} + H_{X} L_{x}^{2} K_{1}^{2} \hfill \\ &\quad + 2 K_{1}^{4} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + 2 \lambda^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} + \bar{g}d_{11} L_{x}^{4} \varOmega K_{1}^{4} + 2 d_{12} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} \hfill \\ &\quad + 4 d_{66} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} + \lambda^{2} e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{2}^{2} - \overline{{M_{5} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} - \lambda^{4} e_{x}^{2} \overline{{M_{5} }} L_{x}^{2} \varOmega^{2} K_{2}^{4} \hfill \\ &\quad - H_{Y} L_{x}^{2} K_{1}^{2} + d_{11} L_{x}^{4} K_{1}^{4} + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} \hfill \\ &\quad - 2 e_{x}^{2} \overline{{M_{5} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} K_{1}^{2} + 2 e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} K_{1}^{2} + \frac{{\bar{g}o_{22} L_{x}^{4} \lambda^{4} \varOmega K_{2}^{4} }}{{\pi^{2} }} \hfill \\ &\quad + 4 \bar{g}d_{66} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} + 2 \bar{g}d_{12} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} + \frac{1}{12}\lambda^{6} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{6} \hfill \\ &\quad + \lambda^{4} e_{x}^{2} H_{Y} L_{x}^{2} K_{2}^{4} + \frac{1}{12} H_{Y} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} - \lambda^{4} e_{x}^{2} H_{X} L_{x}^{2} K_{2}^{4} + \frac{1}{12} H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} \hfill \\ &\quad - W_{elastic \ medium} , \hfill \\ \end{aligned}$$

(83)

$$\begin{aligned} L_{44} &= 2 \frac{{\bar{g}o_{11} L_{x}^{4} \varOmega K_{1}^{4} }}{{\pi^{2} }} + 8 \frac{{o_{66} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} + \bar{g}d_{22} L_{x}^{4} \lambda^{4} \varOmega K_{2}^{4} + 4 \frac{{o_{12} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} \hfill \\ &\quad + 4 \frac{{l_{66} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{4} }} + \frac{{\bar{g}l_{11} L_{x}^{4} \varOmega K_{1}^{4} }}{{\pi^{4} }} + 2 \frac{{l_{12} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{4} }} + \bar{g}k_{44} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} \hfill \\ &\quad + \lambda^{4} e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{2}^{4} - 2 \lambda^{4} e_{x}^{2} \overline{{M_{5} }} L_{x}^{2} \varOmega^{2} K_{2}^{4} + 2 \lambda^{4} K_{2}^{4} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} \hfill \\ &\quad + \frac{1}{6}\lambda^{2} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{2} K_{1}^{4} + \lambda^{4} e_{x}^{2} \overline{{M_{4} }} L_{x}^{2} \varOmega^{2} K_{2}^{4} - 4 \frac{{\lambda^{2} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{2} K_{1}^{4} }}{{\pi^{3} }} \hfill \\ &\quad + \frac{1}{12}e_{x}^{2} H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{4} + \frac{1}{12}\lambda^{4} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{4} K_{1}^{2} + 4 \frac{{\bar{g}o_{12} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} \hfill \\ &\quad + 8 \frac{{\bar{g}o_{66} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} }}{{\pi^{2} }} + 4 \frac{{\bar{g}l_{66} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} }}{{\pi^{4} }} + 2 \frac{{\bar{g}l_{12} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} }}{{\pi^{4} }} \hfill \\ &\quad - 4 \frac{{\lambda^{4} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{4} K_{1}^{2} }}{{\pi^{3} }} - 2 \frac{{\lambda^{4} e_{x}^{2} H_{Y} L_{x}^{4} K_{2}^{4} K_{1}^{2} }}{{\pi^{3} }} + 2 \frac{{\lambda^{2} e_{x}^{2} L_{x}^{2} K_{2}^{2} K_{1}^{2} H_{Y} }}{\pi } \hfill \\ &\quad + 2 \frac{{e_{x}^{2} L_{x}^{2} \lambda^{2} K_{2}^{2} K_{1}^{2} H_{X} }}{\pi } - 2 \frac{{e_{x}^{2} H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{4} }}{{\pi^{3} }} + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} \hfill \\ &\quad + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} + 2 e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} K_{1}^{2} - 4 e_{x}^{2} \overline{{M_{5} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} K_{1}^{2} \hfill \\ &\quad + 2 e_{x}^{2} \overline{{M_{4} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} K_{1}^{2} + 2 \bar{g}d_{12} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} + \frac{{\bar{g}l_{22} L_{x}^{4} \lambda^{4} \varOmega K_{2}^{4} }}{{\pi^{4} }} \hfill \\ &\quad + 4 \bar{g}d_{66} L_{x}^{4} \lambda^{2} \varOmega K_{2}^{2} K_{1}^{2} + k_{55} L_{x}^{2} K_{1}^{2} + 4 d_{66} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} + 2 \frac{{o_{22} L_{x}^{4} \lambda^{4} K_{2}^{4} }}{{\pi^{2} }} \hfill \\ &\quad + \bar{g}k_{55} L_{x}^{2} \varOmega K_{1}^{2} + 2 d_{12} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} + \bar{g}d_{11} L_{x}^{4} \varOmega K_{1}^{4} - 2 \frac{{\lambda^{6} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{6} }}{{\pi^{3} }} \hfill \\ &\quad - 2 \frac{{H_{Y} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{3} }} + 2 \frac{{\lambda^{4} e_{x}^{2} L_{x}^{2} K_{2}^{4} H_{X} }}{\pi } + \frac{1}{6}\lambda^{4} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{4} K_{1}^{2} + d_{11} L_{x}^{4} K_{1}^{4} \hfill \\ &\quad + \frac{{2e_{x}^{2} L_{x}^{2} K_{1}^{4} H_{Y} }}{\pi } + \frac{{2 L_{x}^{2} \lambda^{2} K_{2}^{2} H_{X} }}{\pi } - \frac{{2H_{X} L_{x}^{4} \lambda^{4} K_{2}^{4} }}{{\pi^{3} }} - \frac{{2e_{x}^{2} H_{Y} L_{x}^{4} K_{1}^{6} }}{{\pi^{3} }} \hfill \\ &\quad + \frac{1}{12}\lambda^{6} e_{x}^{2} H_{X} L_{x}^{4} K_{2}^{6} + \lambda^{4} e_{x}^{2} H_{Y} L_{x}^{2} K_{2}^{4} + \frac{1}{12} H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} - H_{Y} L_{x}^{2} K_{1}^{2} \hfill \\ &\quad + \frac{1}{12}H_{Y} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} + 2 K_{1}^{4} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + 2 \lambda^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} + H_{X} L_{x}^{2} K_{1}^{2} \hfill \\ &\quad + 2 \frac{{\bar{g}o_{22} L_{x}^{4} \lambda^{4} \varOmega K_{2}^{4} }}{{\pi^{2} }} - 2 \frac{{H_{X} L_{x}^{4} \lambda^{2} K_{2}^{2} K_{1}^{2} }}{{\pi^{3} }} - \lambda^{4} e_{x}^{2} L_{x}^{2} K_{2}^{4} H_{X} + k_{44} L_{x}^{2} \lambda^{2} K_{2}^{2} \hfill \\ &\quad + \frac{1}{12} H_{Y} L_{x}^{4} K_{1}^{4} + 2 \frac{{H_{Y} L_{x}^{2} K_{1}^{2} }}{\pi } - 2 \frac{{H_{Y} L_{x}^{4} K_{1}^{4} }}{{\pi^{3} }} + H_{Y} L_{x}^{2} \lambda^{2} K_{2}^{2} + e_{x}^{2} H_{X} L_{x}^{2} K_{1}^{4} \hfill \\ &\quad + \frac{1}{12} H_{X} L_{x}^{4} \lambda^{4} K_{2}^{4} + \frac{1}{12}e_{x}^{2} H_{Y} L_{x}^{4} K_{1}^{6} - e_{x}^{2} L_{x}^{2} K_{1}^{4} H_{Y} - L_{x}^{2} \lambda^{2} K_{2}^{2} H_{X} \hfill \\ &\quad + 2 K_{1}^{2} L_{x}^{2} V_{0} \zeta_{31} - 2 \overline{{M_{5} }} L_{x}^{2} \varOmega^{2} K_{1}^{2} + \overline{{M_{4} }} L_{x}^{2} \varOmega^{2} K_{1}^{2} + \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{1}^{2} \hfill \\ &\quad + \frac{{l_{11} L_{x}^{4} K_{1}^{4} }}{{\pi^{4} }} + 2 \frac{{o_{11} L_{x}^{4} K_{1}^{4} }}{{\pi^{2} }} + d_{22} L_{x}^{4} \lambda^{4} K_{2}^{4} + \overline{{M_{2} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} + \overline{{M_{4} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} \hfill \\ &\quad + e_{x}^{2} \overline{{M_{2} }} L_{x}^{2} \varOmega^{2} K_{1}^{4} - 2 \overline{{M_{5} }} L_{x}^{2} \lambda^{2} \varOmega^{2} K_{2}^{2} + e_{x}^{2} \overline{{M_{4} }} L_{x}^{2} \varOmega^{2} K_{1}^{4} - 2 e_{x}^{2} \overline{{M_{5} }} L_{x}^{2} \varOmega^{2} K_{1}^{4} \hfill \\ &\quad + \lambda^{2} e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{2}^{2} + \frac{{l_{22} L_{x}^{4} \lambda^{4} K_{2}^{4} }}{{\pi^{4} }} + \overline{{M_{0} }} \varOmega^{2} + e_{x}^{2} \overline{{M_{0} }} \varOmega^{2} K_{1}^{2} - W_{elastic \ medium} , \hfill \\ \end{aligned}$$

(84)

$$\begin{aligned} L_{45} &= o_{23} L_{x}^{2} \lambda^{2} K_{2}^{2} + \frac{{l_{13} L_{x}^{2} K_{1}^{2} }}{{\pi^{2} }} + k_{44} L_{x}^{2} \lambda^{2} K_{2}^{2} + e_{x}^{2} \overline{{M_{6} }} \varOmega^{2} K_{1}^{2} + 2 K_{1}^{2} L_{x}^{2} V_{0} \zeta_{31} \hfill \\ &\quad + 2 e_{x}^{2} \pi K_{1}^{2} H_{X} - 2 \frac{{K_{1}^{2} L_{x}^{2} H_{Y} }}{\pi } + 2 \frac{{K_{1}^{2} L_{x}^{2} H_{X} }}{\pi } + 2 e_{x}^{2} \pi K_{1}^{2} H_{Y} + \frac{{\bar{g}l_{13} L_{x}^{2} \varOmega K_{1}^{2} }}{{\pi^{2} }} \hfill \\ &\quad + \bar{g}o_{23} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} + \bar{g}k_{44} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} + 2 \lambda^{4} K_{2}^{4} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} - 2 \frac{{\lambda^{4} e_{x}^{2} L_{x}^{2} K_{2}^{4} H_{X} }}{\pi } \hfill \\ &\quad + 2 \frac{{\lambda^{4} e_{x}^{2} L_{x}^{2} K_{2}^{4} H_{Y} }}{\pi }2 \pi H_{Y} + 2 \pi H_{X} + \overline{{M_{6} }} \varOmega^{2} - 2 \frac{{L_{x}^{2} \lambda^{2} K_{2}^{2} H_{X} }}{\pi } + 2 \frac{{L_{x}^{2} \lambda^{2} K_{2}^{2} H_{Y} }}{\pi } \hfill \\ &\quad + 2 \lambda^{2} e_{x}^{2} \pi K_{2}^{2} H_{X} + 2 \lambda^{2} e_{x}^{2} \pi K_{2}^{2} H_{Y} - 2 \frac{{e_{x}^{2} L_{x}^{2} K_{1}^{4} H_{Y} }}{\pi } + 2 \frac{{e_{x}^{2} L_{x}^{2} K_{1}^{4} H_{X} }}{\pi } \hfill \\ &\quad + 2 K_{1}^{4} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + k_{55} L_{x}^{2} K_{1}^{2} + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} + \frac{{\bar{g}l_{23} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} }}{{\pi^{2} }} \hfill \\ &\quad + 2 \lambda^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} + \lambda^{2} e_{x}^{2} \overline{{M_{6} }} \varOmega^{2} K_{2}^{2} + \bar{g}o_{13} L_{x}^{2} \varOmega K_{1}^{2} + \frac{{l_{23} L_{x}^{2} \lambda^{2} K_{2}^{2} }}{{\pi^{2} }} + o_{13} L_{x}^{2} K_{1}^{2} \hfill \\ &\quad + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + \bar{g}k_{55} L_{x}^{2} \varOmega K_{1}^{2} , \hfill \\ \end{aligned}$$

(85)

$$\begin{aligned} L_{46} &= - \frac{1}{2} \zeta_{32} L_{x}^{2} \lambda^{2} K_{2}^{2} - \frac{1}{2} \zeta_{24} L_{x}^{2} \lambda^{2} K_{2}^{2} + 2 \frac{{\zeta_{31} L_{x}^{2} K_{1}^{2} }}{\pi } + 2 \frac{{\bar{g}\zeta_{32} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} }}{\pi } \hfill \\ &\quad - \frac{1}{2} \zeta_{15} L_{x}^{2} K_{1}^{2} - \frac{1}{2} \zeta_{31} L_{x}^{2} K_{1}^{2} - \frac{1}{2} \bar{g}\zeta_{31} L_{x}^{2} \varOmega K_{1}^{2} - \frac{1}{2} \bar{g}\zeta_{15} L_{x}^{2} \varOmega K_{1}^{2} \hfill \\ &\quad + 2 \frac{{\zeta_{32} L_{x}^{2} \lambda^{2} K_{2}^{2} }}{\pi } - \frac{1}{2} \bar{g}\zeta_{32} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} - \frac{1}{2} \bar{g}\zeta_{24} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} + 2 \frac{{\bar{g}\zeta_{31} L_{x}^{2} \varOmega K_{1}^{2} }}{\pi }, \hfill \\ \end{aligned}$$

(86)

$$L_{51} = ih_{13} L_{x} K_{1} + i\bar{g}h_{13} L_{x} \varOmega K_{1} ,$$

(87)

$$L_{52} = - i\bar{g}h_{23} L_{x} \lambda \varOmega K_{2} - ih_{23} L_{x} \lambda K_{2} ,$$

(88)

$$\begin{aligned} L_{53} &= o_{23} L_{x}^{2} \lambda^{2} K_{2}^{2} + \bar{g}o_{23} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} + \bar{g}o_{13} L_{x}^{2} \varOmega K_{1}^{2} + o_{13} L_{x}^{2} K_{1}^{2} + \overline{{M_{6} }} \varOmega^{2} \hfill \\ &\quad + e_{x}^{2} \overline{{M_{6} }} \varOmega^{2} K_{1}^{2} + \lambda^{2} e_{x}^{2} \overline{{M_{6} }} \varOmega^{2} K_{2}^{2} + 2 \lambda^{4} K_{2}^{4} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} \hfill \\ &\quad + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} + 2 K_{1}^{4} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} - H_{Y} K_{1}^{4} L_{x}^{2} e_{x}^{2} + 2 \lambda^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} \hfill \\ &\quad - \lambda^{4} H_{X} K_{2}^{4} L_{x}^{2} e_{x}^{2} + \lambda^{4} H_{Y} K_{2}^{4} L_{x}^{2} e_{x}^{2} - H_{Y} K_{1}^{2} L_{x}^{2} + H_{X} K_{1}^{4} L_{x}^{2} e_{x}^{2} + \lambda^{2} H_{Y} K_{2}^{2} L_{x}^{2} \hfill \\ &\quad + 2 K_{1}^{2} L_{x}^{2} V_{0} \zeta_{31} + H_{X} K_{1}^{2} L_{x}^{2} - \lambda^{2} H_{X} K_{2}^{2} L_{x}^{2} , \hfill \\ \end{aligned}$$

(89)

$$\begin{aligned} L_{54} &= \frac{{l_{13} L_{x}^{2} K_{1}^{2} }}{{\pi^{2} }} + k_{44} L_{x}^{2} \lambda^{2} K_{2}^{2} + o_{23} L_{x}^{2} \lambda^{2} K_{2}^{2} + e_{x}^{2} \overline{{M_{6} }} \varOmega^{2} K_{1}^{2} + 2 K_{1}^{2} L_{x}^{2} V_{0} \zeta_{31} \hfill \\ &\quad + e_{x}^{2} H_{X} L_{x}^{2} K_{1}^{4} - e_{x}^{2} H_{Y} L_{x}^{2} K_{1}^{4} + 2 \frac{{L_{x}^{2} K_{1}^{2} H_{Y} }}{\pi } - H_{X} L_{x}^{2} \lambda^{2} K_{2}^{2} + H_{Y} L_{x}^{2} \lambda^{2} K_{2}^{2} \hfill \\ &\quad + \bar{g}k_{44} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} + \bar{g}o_{23} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} + \frac{{\bar{g}l_{13} L_{x}^{2} \varOmega K_{1}^{2} }}{{\pi^{2} }} + \lambda^{2} e_{x}^{2} \overline{{M_{6} }} \varOmega^{2} K_{2}^{2} \hfill \\ &\quad + 2 \lambda^{4} K_{2}^{4} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} + 2 \frac{{\lambda^{4} e_{x}^{2} L_{x}^{2} K_{2}^{4} H_{X} }}{\pi } + 2 \frac{{H_{X} L_{x}^{2} \lambda^{2} K_{2}^{2} }}{\pi } + \lambda^{4} e_{x}^{2} H_{Y} L_{x}^{2} K_{2}^{4} \hfill \\ &\quad + 2 \frac{{e_{x}^{2} H_{Y} L_{x}^{2} K_{1}^{4} }}{\pi } - \lambda^{4} e_{x}^{2} L_{x}^{2} K_{2}^{4} H_{X} + 2 K_{1}^{4} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + 2 \lambda^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} \hfill \\ &\quad + \bar{g}k_{55} L_{x}^{2} \varOmega K_{1}^{2} + \bar{g}o_{13} L_{x}^{2} \varOmega K_{1}^{2} + \overline{{M_{6} }} \varOmega^{2} - L_{x}^{2} K_{1}^{2} H_{Y} + k_{55} L_{x}^{2} K_{1}^{2} + o_{13} L_{x}^{2} K_{1}^{2} \hfill \\ &\quad + \frac{{l_{23} L_{x}^{2} \lambda^{2} K_{2}^{2} }}{{\pi^{2} }} + H_{X} L_{x}^{2} K_{1}^{2} + 2 \frac{{e_{x}^{2} L_{x}^{2} \lambda^{2} K_{2}^{2} K_{1}^{2} H_{X} }}{\pi } + 2 \frac{{\lambda^{2} e_{x}^{2} L_{x}^{2} K_{2}^{2} K_{1}^{2} H_{Y} }}{\pi } \hfill \\ &\quad + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} + \frac{{\bar{g}l_{23} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} }}{{\pi^{2} }} + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} , \hfill \\ \end{aligned}$$

(90)

$$\begin{aligned} L_{55} = k_{44} L_{x}^{2} \lambda^{2} K_{2}^{2} + e_{x}^{2} \overline{{M_{7} }} \varOmega^{2} K_{1}^{2} + 2 K_{1}^{2} L_{x}^{2} V_{0} \zeta_{31} + 2 e_{x}^{2} \pi K_{1}^{2} H_{X} + 2 e_{x}^{2} \pi K_{1}^{2} H_{Y} \hfill \\ &\quad + 2 \frac{{L_{x}^{2} K_{1}^{2} H_{X} }}{\pi } + \bar{g}k_{44} L_{x}^{2} \lambda^{2} \varOmega K_{2}^{2} + 2 \lambda^{4} K_{2}^{4} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} + 2 \frac{{\lambda^{4} e_{x}^{2} L_{x}^{2} K_{2}^{4} H_{Y} }}{\pi } \hfill \\ &\quad + 2 \lambda^{2} e_{x}^{2} \pi K_{2}^{2} H_{Y} + 2 \frac{{e_{x}^{2} L_{x}^{2} K_{1}^{4} H_{X} }}{\pi } + 2 \frac{{L_{x}^{2} \lambda^{2} K_{2}^{2} H_{Y} }}{\pi } + 2 \frac{{\lambda^{2} e_{x}^{2} L_{x}^{2} K_{2}^{2} K_{1}^{2} H_{Y} }}{\pi } \hfill \\ &\quad + 2 \lambda^{2} e_{x}^{2} \pi K_{2}^{2} H_{X} + 2 K_{1}^{4} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + 2 \lambda^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} + \lambda^{2} e_{x}^{2} \overline{{M_{7} }} \varOmega^{2} K_{2}^{2} + l_{33} \hfill \\ &\quad + \bar{g}k_{55} L_{x}^{2} \varOmega K_{1}^{2} + 2 \pi H_{X} + 2 \pi H_{Y} + \overline{{M_{7} }} \varOmega^{2} + \bar{g}l_{33} \varOmega + k_{55} L_{x}^{2} K_{1}^{2} \hfill \\ &\quad + 2 \frac{{e_{x}^{2} L_{x}^{2} \lambda^{2} K_{2}^{2} K_{1}^{2} H_{X} }}{\pi } + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{31} e_{x}^{2} + 2 \lambda^{2} K_{1}^{2} K_{2}^{2} L_{x}^{2} V_{0} \zeta_{32} e_{x}^{2} , \hfill \\ \end{aligned} \hfill \\$$

(91)

$$\begin{aligned} L_{56} &= - \frac{1}{2}\bar{g}\zeta_{24} \varOmega K_{2}^{2} L_{x}^{2} \lambda^{2} - \frac{1}{2}\zeta_{24} K_{2}^{2} L_{x}^{2} \lambda^{2} - \frac{1}{2}\bar{g}\varOmega \pi^{2} \zeta_{33} - \frac{1}{2} \pi^{2} \zeta_{33} - \frac{1}{2} K_{1}^{2} \zeta_{15} L_{x}^{2} \hfill \\ &\quad - \frac{1}{2} \bar{g}\varOmega K_{1}^{2} \zeta_{15} L_{x}^{2} , \hfill \\ \end{aligned}$$

(92)

$$L_{61} = 0 ,$$

(93)

$$L_{62} = 0 ,$$

(94)

$$L_{63} = 2 \frac{{\zeta_{31} L_{x}^{2} K_{1}^{2} }}{\pi } + 2 \frac{{\zeta_{32} L_{x}^{2} \lambda^{2} K_{2}^{2} }}{\pi } ,$$

(95)

$$\begin{aligned} L_{64} &= - \frac{1}{2}\zeta_{15} L_{x}^{2} K_{1}^{2} - \frac{1}{2}\zeta_{24} L_{x}^{2} \lambda^{2} K_{2}^{2} - \frac{1}{2}\zeta_{31} L_{x}^{2} K_{1}^{2} + 2 \frac{{\zeta_{32} L_{x}^{2} \lambda^{2} K_{2}^{2} }}{\pi } \hfill \\ &\quad + 2 \frac{{\zeta_{31} L_{x}^{2} K_{1}^{2} }}{\pi } - \frac{1}{2} \zeta_{32} L_{x}^{2} \lambda^{2} K_{2}^{2} , \hfill \\ \end{aligned}$$

(96)

$$L_{65} = - \frac{1}{2} \zeta_{15} L_{x}^{2} K_{1}^{2} - \frac{1}{2} \zeta_{24} L_{x}^{2} \lambda^{2} K_{2}^{2} - \frac{1}{2}\pi^{2} \zeta_{33} ,$$

(97)

$$L_{66} = - \frac{1}{2} \vartheta_{11} K_{1}^{2} L_{x}^{2} - \frac{1}{2} \vartheta_{22} K_{2}^{2} L_{x}^{2} \lambda^{2} - \frac{1}{2} \pi^{2} \vartheta_{33} ,$$

(98)

where $W_{elastic \ medium}$ in Eqs. (75), (76), (82) and (83) are

$$W_{elastic \ medium} = j \left( {q_{elastic \ medium} } \right) ,$$

(99)

$$\begin{aligned} q_{elastic \ medium} &= \left( { - \lambda^{4} G_{F2} K_{2}^{4} L_{x}^{2} e_{x}^{2} - \lambda^{2} G_{F1} K_{1}^{2} K_{2}^{2} L_{x}^{2} e_{x}^{2} - \lambda^{2} G_{F2} K_{1}^{2} K_{2}^{2} L_{x}^{2} e_{x}^{2} } \right. \hfill \\ &\quad \left. { - G_{F1} K_{1}^{4} L_{x}^{2} e_{x}^{2} - \lambda^{2} G_{F2} K_{2}^{2} L_{x}^{2} - G_{F1} K_{1}^{2} L_{x}^{2} } \right)cos\theta^{2} + \left( { - \lambda^{4} G_{F1} K_{2}^{4} L_{x}^{2} e_{x}^{2} } \right. \hfill \\ &\quad - \lambda^{2} G_{F1} K_{1}^{2} K_{2}^{2} L_{x}^{2} e_{x}^{2} - \lambda^{2} G_{F2} K_{1}^{2} K_{2}^{2} L_{x}^{2} e_{x}^{2} - G_{F2} K_{1}^{4} L_{x}^{2} e_{x}^{2} - G_{F2} K_{1}^{2} L_{x}^{2} \hfill \\ &\quad \left. { - \lambda^{2} G_{F1} K_{2}^{2} L_{x}^{2} } \right)sin\theta^{2} + \left( {2 \lambda^{3} G_{F1} K_{1} K_{2}^{3} L_{x}^{2} e_{x}^{2} - 2 \lambda^{3} G_{F2} K_{1} K_{2}^{3} L_{x}^{2} e_{x}^{2} } \right. \hfill \\ &\quad + 2 \lambda G_{F1} K_{1}^{3} K_{2} L_{x}^{2} e_{x}^{2} - 2 \lambda G_{F2} K_{1}^{3} K_{2} L_{x}^{2} e_{x}^{2} + 2 \lambda G_{F1} K_{1} K_{2} L_{x}^{2} \hfill \\ &\quad \left. { - 2 \lambda G_{F2} K_{1} K_{2} L_{x}^{2} } \right)cos\theta sin\theta - K_{W} - K_{1}^{2} K_{W} e_{x}^{2} - \lambda^{2} K_{2}^{2} K_{W} e_{x}^{2} - \varOmega C_{d} \hfill \\ &\quad - \varOmega C_{d} K_{1}^{2} e_{x}^{2} - \varOmega \lambda^{2} C_{d} K_{2}^{2} e_{x}^{2} , \hfill \\ \end{aligned}$$

(100)

in which $j$ for wave propagation types such as in-phase, one micro plate fixed and out-of-phase are equal 1, 2 and 3, respectively.