Nonlinear Vibration Analysis of Laminated Magneto-Electro-Elastic Rectangular Plate Based on Third-Order Shear Deformation Theory

Abstract

Nonlinear vibration of magneto-electro-elastic rectangular thick plate is studied based on the third-order shear deformation theory. The plate is symmetrically laminated and simply supported along all of its edges. The magneto-electric boundary condition on upper and bottom surfaces of the plate is considered to be closed-circuit. Gauss’s laws for electrostatics and magnetostatics are used to model the electric and magnetic behavior of the plate. After deriving the nonlinear partial differential equations of motion, Galerkin method is applied to transform these equations into a single ordinary differential equation. This equation is solved analytically by using Lindstedt–Poincaré and multiple time scales methods. Several examples are provided to investigate the effects of different parameters on the nonlinear vibration of these plates.

Appendices

Appendix 1

$$\begin{aligned} & C_{11} \left( {\frac{{\partial^{2} u_{0} }}{{\partial x^{2} }} + \frac{{\partial w_{0} }}{\partial x}\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right)h + C_{66} \left( {\frac{{\partial^{2} u_{0} }}{{\partial y^{2} }} + \frac{{\partial^{2} v_{0} }}{\partial x\partial y} + \frac{{\partial w_{0} }}{\partial y}\frac{{\partial^{2} w_{0} }}{\partial x\partial y} + \frac{{\partial w_{0} }}{\partial x}\frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right)h \\ & + C_{12} \left( {\frac{{\partial^{2} v_{0} }}{\partial x\partial y} + \frac{{\partial w_{0} }}{\partial y}\frac{{\partial^{2} w_{0} }}{\partial x\partial y}} \right)h + \frac{2}{3}\frac{{e_{31}^{2} }}{{\eta_{33} }}\left( {2\frac{{\partial w_{0} }}{\partial x}\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }} + 2\frac{{\partial w_{0} }}{\partial y}\frac{{\partial^{2} w_{0} }}{\partial x\partial y} + 2\frac{{\partial^{2} u_{0} }}{{\partial x^{2} }} + 2\frac{{\partial^{2} v_{0} }}{\partial x\partial y}} \right)h \\ & \frac{2}{3}\frac{{q_{31}^{2} }}{{\mu_{33} }}\left( {2\frac{{\partial w_{0} }}{\partial x}\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }} + 2\frac{{\partial w_{0} }}{\partial y}\frac{{\partial^{2} w_{0} }}{\partial x\partial y} + 2\frac{{\partial^{2} u_{0} }}{{\partial x^{2} }} + 2\frac{{\partial^{2} v_{0} }}{\partial x\partial y}} \right)h = 0 \\ \end{aligned}$$

(54)

$$\begin{aligned} & C_{22} \left( {\frac{{\partial^{2} v_{0} }}{{\partial y^{2} }} + \frac{{\partial w_{0} }}{\partial y}\frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right)h + C_{66} \left( {\frac{{\partial^{2} u_{0} }}{\partial x\partial y} + \frac{{\partial^{2} v_{0} }}{{\partial x^{2} }} + \frac{{\partial w_{0} }}{\partial y}\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }} + \frac{{\partial w_{0} }}{\partial x}\frac{{\partial^{2} w_{0} }}{\partial y\partial x}} \right)h \\ & + C_{12} \left( {\frac{{\partial^{2} u_{0} }}{\partial x\partial y} + \frac{{\partial w_{0} }}{\partial x}\frac{{\partial^{2} w_{0} }}{\partial x\partial y}} \right)h + \frac{2}{3}\frac{{e_{31}^{2} }}{{\eta_{33} }}\left( {2\frac{{\partial w_{0} }}{\partial x}\frac{{\partial^{2} w_{0} }}{\partial x\partial y} + 2\frac{{\partial w_{0} }}{\partial y}\frac{{\partial^{2} w_{0} }}{{\partial y^{2} }} + 2\frac{{\partial^{2} u_{0} }}{\partial x\partial y} + 2\frac{{\partial^{2} v_{0} }}{{\partial y^{2} }}} \right)h \\ & \frac{2}{3}\frac{{q_{31}^{2} }}{{\mu_{33} }}\left( {2\frac{{\partial w_{0} }}{\partial x}\frac{{\partial^{2} w_{0} }}{\partial x\partial y} + 2\frac{{\partial w_{0} }}{\partial y}\frac{{\partial^{2} w_{0} }}{{\partial y^{2} }} + 2\frac{{\partial^{2} u_{0} }}{\partial x\partial y} + 2\frac{{\partial^{2} v_{0} }}{{\partial y^{2} }}} \right)h = 0 \\ \end{aligned}$$

(55)

$$\begin{aligned} & \frac{8}{15}C_{55} h\left( {\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right) + \frac{8}{15}C_{55} h\left( {\frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right) - \frac{{h^{3} }}{252}\left( {C_{11} \left( {\frac{{\partial^{3} \phi_{x} }}{{\partial x^{3} }} + \frac{{\partial^{4} w_{0} }}{{\partial x^{4} }}} \right) + C_{12} \left( {\frac{{\partial^{3} \phi_{y} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{4} w_{0} }}{{\partial x^{2} \partial y^{2} }}} \right)} \right) \\ & - \;\frac{{h^{5} }}{252}\frac{{e_{31} }}{{\eta_{33} }}\left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{4} w_{0} }}{{\partial x^{4} }} + \frac{{\partial^{4} w_{0} }}{{\partial x^{2} \partial y^{2} }} + \frac{{\partial^{3} \phi_{x} }}{{\partial x^{3} }} + \frac{{\partial^{3} \phi_{y} }}{{\partial x^{2} \partial y}}} \right) \\ & - \;\frac{{h^{5} }}{252}\frac{{q_{31} }}{{\mu_{33} }}\left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{4} w_{0} }}{{\partial x^{4} }} + \frac{{\partial^{4} w_{0} }}{{\partial x^{2} \partial y^{2} }} + \frac{{\partial^{3} \phi_{x} }}{{\partial x^{3} }} + \frac{{\partial^{3} \phi_{y} }}{{\partial x^{2} \partial y}}} \right) + \frac{{h^{3} }}{60}\left( {C_{11} \frac{{\partial^{3} \phi_{x} }}{{\partial x^{3} }} + C_{12} \frac{{\partial^{3} \phi_{y} }}{{\partial y\partial x^{2} }}} \right) \\ & + \;\frac{{e_{31} }}{{60\eta_{33} }}\left( {e_{15} \left( {\frac{{\partial^{4} w_{0} }}{{\partial x^{4} }} + \frac{{\partial^{4} w_{0} }}{{\partial x^{2} \partial y^{2} }}} \right) + \left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{3} \phi_{x} }}{{\partial x^{3} }} + \frac{{\partial^{3} \phi_{y} }}{{\partial x^{2} \partial y}}} \right)} \right)h^{3} \\ & + \;\frac{{q_{31} }}{{60\mu_{33} }}\left( {q_{15} \left( {\frac{{\partial^{4} w_{0} }}{{\partial x^{4} }} + \frac{{\partial^{4} w_{0} }}{{\partial x^{2} \partial y^{2} }}} \right) + \left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{3} \phi_{x} }}{{\partial x^{3} }} + \frac{{\partial^{3} \phi_{y} }}{{\partial x^{2} \partial y}}} \right)} \right)h^{3} \\ & + \;\frac{{h^{3} }}{126}C_{66} \left( {\frac{{\partial^{3} \phi_{x} }}{{\partial y^{2} \partial x}} + \frac{{\partial^{3} \phi_{y} }}{{\partial x^{2} \partial y}} + 2\frac{{\partial^{4} w_{0} }}{{\partial x^{2} \partial y^{2} }}} \right) + \frac{{h^{3} }}{30}C_{66} \left( {\frac{{\partial^{3} \phi_{x} }}{{\partial y^{2} \partial x}} + \frac{{\partial^{3} \phi_{y} }}{{\partial x^{2} \partial y}}} \right) \\ & - \;\frac{{h^{3} }}{252}\left( {C_{12} \left( {\frac{{\partial^{3} \phi_{x} }}{{\partial y^{2} \partial x}} + \frac{{\partial^{4} w_{0} }}{{\partial x^{2} \partial y^{2} }}} \right) + C_{22} \left( {\frac{{\partial^{3} \phi_{y} }}{{\partial y^{3} }} + \frac{{\partial^{4} w_{0} }}{{\partial y^{4} }}} \right)} \right) \\ & - \;\frac{{h^{5} e_{31} }}{{252\eta_{33} }}\left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{4} w_{0} }}{{\partial y^{4} }} + \frac{{\partial^{4} w_{0} }}{{\partial x^{2} \partial y^{2} }} + \frac{{\partial^{3} \phi_{y} }}{{\partial y^{3} }} + \frac{{\partial^{3} \phi_{x} }}{{\partial y^{2} \partial x}}} \right) \\ & - \;\frac{{h^{5} q_{31} }}{{252\mu_{33} }}\left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{4} w_{0} }}{{\partial y^{4} }} + \frac{{\partial^{4} w_{0} }}{{\partial x^{2} \partial y^{2} }} + \frac{{\partial^{3} \phi_{y} }}{{\partial y^{3} }} + \frac{{\partial^{3} \phi_{x} }}{{\partial y^{2} \partial x}}} \right) + \frac{{h^{3} }}{60}\left( {C_{12} \frac{{\partial^{3} \phi_{x} }}{{\partial y^{2} \partial x}} + C_{22} \frac{{\partial^{3} \phi_{y} }}{{\partial y^{3} }}} \right) \\ & + \;\frac{{e_{31} }}{{60\eta_{33} }}\left( {e_{15} \left( {\frac{{\partial^{4} w_{0} }}{{\partial x^{2} \partial y^{2} }} + \frac{{\partial^{4} w_{0} }}{{\partial y^{4} }}} \right) + \left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{3} \phi_{y} }}{{\partial y^{3} }} + \frac{{\partial^{3} \phi_{x} }}{{\partial y^{2} \partial x}}} \right)} \right)h^{3} \\ & + \;\frac{{q_{31} }}{{60\mu_{33} }}\left( {q_{15} \left( {\frac{{\partial^{4} w_{0} }}{{\partial x^{4} }} + \frac{{\partial^{4} w_{0} }}{{\partial x^{2} \partial y^{2} }}} \right) + \left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{3} \phi_{y} }}{{\partial y^{3} }} + \frac{{\partial^{3} \phi_{x} }}{{\partial y^{2} \partial x}}} \right)} \right)h^{3} - \rho hw_{tt} + q_{0} \cos \left( {\varOmega \,t} \right) = 0 \\ \end{aligned}$$

(56)

where ρ,$q_{0}$ and Ω are density, initial load and exciting frequency.

$$\begin{aligned} & \frac{{h^{3} }}{252}\left( {C_{11} \left( {\frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{3} w_{0} }}{{\partial x^{3} }}} \right) + C_{12} \left( {\frac{{\partial^{2} \phi_{y} }}{\partial x\partial y} + \frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }}} \right)} \right) \\ & + \;\frac{{h^{5} }}{252}\frac{{e_{31} }}{{\eta_{33} }}\left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{3} w_{0} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}} \right) \\ & + \;\frac{{h^{5} }}{252}\frac{{q_{31} }}{{\mu_{33} }}\left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{3} w_{0} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}} \right) \\ & + \;\frac{{h^{3} }}{60}\left( {C_{11} \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + C_{12} \frac{{\partial^{2} \phi_{y} }}{\partial y\partial x}} \right) \\ & + \;\frac{{e_{31} }}{{60\eta_{33} }}\left( {e_{15} \left( {\frac{{\partial^{3} w_{0} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }}} \right) + \left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}} \right)} \right)h^{3} \\ & + \;\frac{{q_{31} }}{{60\mu_{33} }}\left( {q_{15} \left( {\frac{{\partial^{3} w_{0} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }}} \right) + \left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}} \right)} \right)h^{3} \\ & - \;\frac{{h^{3} C_{11} }}{60}\left( {\frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{3} w_{0} }}{{\partial x^{3} }}} \right) - \frac{{h^{3} C_{12} }}{60}\left( {\frac{{\partial^{2} \phi_{y} }}{\partial x\partial y} + \frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }}} \right) \\ & + \;\frac{{e_{31} }}{{60\eta_{33} }}\left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{3} w_{0} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}} \right)h^{3} \\ & + \;\frac{{q_{31} }}{{60\mu_{33} }}\left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{3} w_{0} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}} \right)h^{3} \\ & + \;\frac{{h^{3} }}{12}\left( {C_{11} \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + C_{12} \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}} \right) \\ & + \;\frac{{e_{31} }}{{12\eta_{33} }}\left( {e_{15} \left( {\frac{{\partial^{3} w_{0} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }}} \right) + \left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}} \right)} \right)h^{3} \\ & + \;\frac{{q_{31} }}{{12\mu_{33} }}\left( {q_{15} \left( {\frac{{\partial^{3} w_{0} }}{{\partial x^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }}} \right) + \left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}} \right)} \right)h^{3} \\ & + \;\frac{{h^{3} C_{66} }}{252}\left( {\frac{{\partial^{2} \phi_{x} }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y} + 2\frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }}} \right) + \frac{{h^{3} C_{66} }}{60}\left( {\frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}} \right) \\ & - \;\frac{{h^{3} C_{66} }}{60}\left( {\frac{{\partial^{2} \phi_{x} }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y} + 2\frac{{\partial^{3} w_{0} }}{{\partial x\partial y^{2} }}} \right) + \frac{{h^{3} C_{66} }}{12}\left( {\frac{{\partial^{2} \phi_{x} }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}} \right) \\ & - \;\frac{8}{15}hC_{55} \left( {\phi_{x} + \frac{{\partial w_{0} }}{\partial x}} \right) = 0 \\ \end{aligned}$$

(57)

$$\begin{aligned} & \frac{{h^{3} }}{252}\left( {C_{12} \left( {\frac{{\partial^{2} \phi_{x} }}{\partial x\partial y} + \frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}}} \right) + C_{22} \left( {\frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + \frac{{\partial^{3} w_{0} }}{{\partial y^{3} }}} \right)} \right) \\ & + \;\frac{{h^{5} }}{252}\frac{{e_{31} }}{{\eta_{33} }}\left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{3} w_{0} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}} \right) \\ & + \;\frac{{h^{5} }}{252}\frac{{q_{31} }}{{\mu_{33} }}\left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{3} w_{0} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}} \right) \\ & + \;\frac{{h^{3} }}{60}\left( {C_{12} \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y} + C_{22} \frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }}} \right) \\ & + \;\frac{{e_{31} }}{{60\eta_{33} }}\left( {e_{15} \left( {\frac{{\partial^{3} w_{0} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}}} \right) + \left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}} \right)} \right)h^{3} \\ & + \;\frac{{q_{31} }}{{60\mu_{33} }}\left( {q_{15} \left( {\frac{{\partial^{3} w_{0} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}}} \right) + \left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}} \right)} \right)h^{3} \\ & - \;\frac{{h^{3} C_{22} }}{60}\left( {\frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + \frac{{\partial^{3} w_{0} }}{{\partial y^{3} }}} \right) - \frac{{h^{3} C_{12} }}{60}\left( {\frac{{\partial^{2} \phi_{x} }}{\partial x\partial y} + \frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}}} \right) \\ & + \;\frac{{e_{31} }}{{60\eta_{33} }}\left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{3} w_{0} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}} \right)h^{3} \\ & + \;\frac{{q_{31} }}{{60\mu_{33} }}\left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{3} w_{0} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}} + \frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}} \right)h^{3} \\ & + \;\frac{{h^{3} }}{12}\left( {C_{12} \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y} + C_{22} \frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }}} \right) \\ & + \;\frac{{e_{31} }}{{12\eta_{33} }}\left( {e_{15} \left( {\frac{{\partial^{3} w_{0} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}}} \right) + \left( {e_{31} + e_{15} } \right)\left( {\frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}} \right)} \right)h^{3} \\ & + \;\frac{{q_{31} }}{{12\mu_{33} }}\left( {q_{15} \left( {\frac{{\partial^{3} w_{0} }}{{\partial y^{3} }} + \frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}}} \right) + \left( {q_{31} + q_{15} } \right)\left( {\frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}} \right)} \right)h^{3} \\ & + \;\frac{{h^{3} C_{66} }}{252}\left( {\frac{{\partial^{2} \phi_{x} }}{\partial y\partial x} + \frac{{\partial^{2} \phi_{y} }}{{\partial x^{2} }} + 2\frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}}} \right) + \frac{{h^{3} C_{66} }}{60}\left( {\frac{{\partial^{2} \phi_{x} }}{\partial x\partial y} + \frac{{\partial^{2} \phi_{y} }}{{\partial x^{2} }}} \right) \\ & - \;\frac{{h^{3} C_{66} }}{60}\left( {\frac{{\partial^{2} \phi_{x} }}{\partial y\partial x} + \frac{{\partial^{2} \phi_{y} }}{{\partial x^{2} }} + 2\frac{{\partial^{3} w_{0} }}{{\partial x^{2} \partial y}}} \right) + \frac{{h^{3} C_{66} }}{12}\left( {\frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}} \right) \\ & - \;\frac{8}{15}hC_{44} \left( {\phi_{y} + \frac{{\partial w_{0} }}{\partial y}} \right) = 0 \\ \end{aligned}$$

(58)

Appendix 2

$$\begin{aligned} S_{1} = & - \frac{2}{27}\frac{1}{{ba^{2} n\eta_{B33} \mu_{F33} }}\left( {\pi^{2} h^{3} m\left( {( - 1)^{n} - 1} \right)} \right.\left( {\left( {\left( {\left( {\frac{3}{8}n^{2} (2C_{66B} + 4C_{66F} + C_{12B} } \right.} \right.} \right.} \right. \\ & \left. { + \;2C_{12F} )a^{2} + \frac{3}{8}b^{2} m^{2} (C_{11B} + 2C_{11F} )} \right)\left. {\mu_{F33} + q_{31}^{2} + (a^{2} n^{2} + b^{2} m^{2} )} \right)\eta_{B33} \\ & \left. { + \;\frac{1}{2}e_{31}^{2} \mu_{F33} (a^{2} n^{2} + b^{2} m^{2} )} \right)\left( {( - 1)^{n} } \right)^{2} + \left( {\left( {\left( {\frac{3}{8}n^{2} (2C_{66B} + 4C_{66F} + C_{12B} } \right.} \right.} \right. \\ & \left. { + \;2C_{12F} )a^{2} + \frac{3}{8}b^{2} m^{2} (C_{11B} + 2C_{11F} )} \right)\left. {\mu_{F33} + q_{31}^{2} + (a^{2} n^{2} + b^{2} m^{2} )} \right)\eta_{B33} \\ & \left. { + \;\frac{1}{2}e_{31}^{2} \mu_{F33} (a^{2} n^{2} + b^{2} m^{2} )} \right)( - 1)^{n} + \left( {\left( {\frac{3}{8}n^{2} (C_{12B} - C_{66B} + 2C_{12F} - 2C_{66F} )a^{2} } \right.} \right. \\ & \left. { - \;\frac{3}{4}b^{2} m^{2} (C_{11B} + 2C_{11F} )} \right)\left. {\mu_{F33} + q_{31}^{2} + (a^{2} n^{2} - 2b^{2} m^{2} )} \right)\eta_{B33} \quad + \frac{1}{2}e_{31}^{2} \mu_{F33} (a^{2} n^{2} \\ & \left. { - \;2b^{2} m^{2} )} \right)\left. {W^{2} } \right) \\ \end{aligned}$$

(59)

$$\begin{aligned} S_{2} = & - \frac{1}{36}\frac{1}{{a\eta_{B33} b^{2} m}}\left( {\pi^{2} n\left( {\frac{1}{3}he_{31}^{2} (a^{2} n^{2} + b^{2} m^{2} )( - 1)^{3m} + 3(m^{2} (A_{12} + 2A_{66} )b^{2} } \right.} \right. \\ & a^{2} n^{2} A_{11} )\eta_{B33} (( - 1)^{m} )^{3} - 9\eta_{B33} (a^{2} n^{2} A_{11} + b^{2} m^{2} A_{66} )( - 1)^{m} + \left( {\left( {( - 1)^{1 + m} } \right.} \right. \\ & \left. {\left. {\left. { + \;\frac{2}{3}} \right)e_{31}^{2} h + 6A_{11} \eta_{B33} } \right)\left. {n^{2} a^{2} - 3b^{2} \left( {\frac{1}{9}he_{31}^{2} + \eta_{B33} (A_{12} + A_{66} )} \right)m^{2} } \right)Wh^{2} } \right) \\ \end{aligned}$$

(60)

$$\begin{aligned} S_{3} = & - \frac{1}{3}\frac{1}{{ab\eta_{B33} \mu_{F33} }}\left( {\left( {\left( {\left( {m^{2} (C_{11B} + 2C_{11F} )b^{2} + \frac{1}{4}a^{2} n^{2} (C_{66B} + 2C_{66F} )} \right)\mu_{F33} } \right.} \right.} \right. \\ & + \left. {\frac{8}{3}b^{2} m^{2} q_{31}^{2} } \right)\left. {\eta_{B33} + \frac{4}{3}\mu_{F33} b^{2} m^{2} e_{31}^{2} } \right)\left. {U\pi^{2} h^{2} } \right) \\ \end{aligned}$$

(61)

$$\begin{aligned} S_{4} = & \frac{8}{27}\frac{1}{{\eta_{B33} \mu_{F33} }}\left( {\left( {\left( {(C_{12B} + C_{66B} + 2C_{12F} + 2C_{66F} )\mu_{F33} + \frac{8}{3}q_{31}^{2} } \right)} \right.} \right.\eta_{B33} \\ & + \left. {\frac{4}{3}e_{31}^{2} \mu_{F33} } \right)(( - 1)^{m} - 1)(( - 1)^{n} - 1)h^{2} \left( {(( - 1)^{m} )^{2} + ( - 1)^{m} + 1} \right)\left( {(( - 1)^{n} )^{2} } \right. \\ & \left. { + \left. {( - 1)^{n} - \frac{1}{2}} \right)V} \right) \\ \end{aligned}$$

(62)

$$\begin{aligned} S_{5} = & - \frac{2}{27}\frac{1}{{\eta_{B33} a\mu_{F33} b^{2} m}}\left( {\left( {\left( {\left( {\left( {\frac{3}{8}m^{2} (2C_{66B} + 4C_{66F} + 2C_{12F} + C_{12B} )b^{2} } \right.} \right.} \right.} \right.} \right. \\ & \left. {\left. { + \;\frac{3}{8}a^{2} n^{2} (C_{22B} + 2C_{22F} )} \right)\mu_{F33} + q_{31}^{2} (a^{2} n^{2} + b^{2} m^{2} )} \right)\eta_{B33} + \frac{1}{2}e_{31}^{2} \mu_{F33} (a^{2} n^{2} \\ & \left. {\left. { + \;b^{2} m^{2} } \right)} \right)(( - 1)^{m} )^{2} + \left( {\left( {\left( {\frac{3}{8}m^{2} (2C_{66B} + 4C_{66F} + 2C_{12F} + C_{12B} )b^{2} } \right.} \right.} \right. \\ & \left. { + \;\frac{3}{8}a^{2} n^{2} (C_{22B} + 2C_{22F} )} \right)\left. {\mu_{F33} + q_{31}^{2} (a^{2} n^{2} + b^{2} m^{2} )} \right)\eta_{B33} + \frac{1}{2}e_{31}^{2} \mu_{F33} \left( {a^{2} n^{2} } \right. \\ & \left. {\left. { + \;b^{2} m^{2} } \right)} \right)( - 1)^{m} + \left( {\left( {\frac{3}{8}m^{2} (C_{12B} - C_{66B} + 2C_{12F} - 2C_{66F} )b^{2} - \frac{3}{4}a^{2} n^{2} (C_{22B} } \right.} \right. \\ & \left. { + \;2C_{22F} )} \right)\left. {\mu_{F33} - 2a^{2} n^{2} q_{31}^{2} + b^{2} m^{2} q_{31}^{2} } \right)\left. {\eta_{B33} - \mu_{F33} (a^{2} n^{2} - \frac{1}{2}b^{2} m^{2} )e_{31}^{2} } \right) \\ & \left. {W^{2} h^{3} n\pi^{2} (( - 1)^{m} - 1)} \right) \\ \end{aligned}$$

(63)

$$\begin{aligned} S_{6} = & - \frac{8}{81}\frac{1}{{\eta_{B33} }}\left( {Wh\left( {\frac{1}{2}he_{31}^{2} ( - 1)^{1 + 3n} + \frac{3}{2}he_{31}^{2} ( - 1)^{3n + m} + \frac{3}{2}he_{31}^{2} ( - 1)^{1 + m} + he_{31}^{2} ( - 1)^{3m} } \right.} \right. \\ & - \;9\eta_{B33} (( - 1)^{m} - 1)\left( {(( - 1)^{m} )^{2} + ( - 1)^{m} - \frac{1}{2}} \right)(A_{12} + A_{66} )(( - 1)^{n} )^{3} \\ & + \;9\eta_{B33} (A_{12} + A_{66} )(( - 1)^{m} )^{3} - \frac{27}{2}\eta_{B33} (A_{12} + A_{66} )( - 1)^{m} + \left( {( - 1)^{1 + 3n + 3m} + \frac{1}{2}} \right) \\ & \left. {\left. {e_{31}^{2} h + \frac{9}{2}\eta_{B33} (A_{12} + A_{66} )} \right)} \right) \\ \end{aligned}$$

(64)

$$\begin{aligned} S_{7} = & \frac{8}{27}\frac{1}{{\eta_{B33} \mu_{F33} }}\left( {\left( {(( - 1)^{m} )^{2} + ( - 1)^{m} - \frac{1}{2}} \right)} \right.\left( {\left( {(C_{12B} + C_{66B} + 2C_{12F} + 2C_{66F} )\mu_{F33} } \right.} \right. \\ & \left. {\frac{8}{3}q_{31}^{2} } \right)\left. {\eta_{B33} + \frac{4}{3}e_{31}^{2} \mu_{F33} } \right)(( - 1)^{m} - 1)\left( {(( - 1)^{n} )^{2} + ( - 1)^{n} } \right. \\ & \left. {\left. { + \;1} \right)(( - 1)^{n} - 1)h^{2} U} \right) \\ \end{aligned}$$

(65)

$$\begin{aligned} S_{8} = & - \frac{1}{3}\frac{1}{{b\eta_{B33} a\mu_{F33} }}\left( {\left( {h^{2} \left( {\left( {\left( {n^{2} (C_{22B} + 2C_{22F} )a^{2} + \frac{1}{4}b^{2} m^{2} (C_{66B} + 2C_{66F} )} \right)} \right.\mu_{F33} } \right.} \right.} \right. \\ & \left. {\left. {\left. { + \frac{8}{3}a^{2} n^{2} q_{31}^{2} } \right)\eta_{B33} + \frac{4}{3}\mu_{F33} a^{2} n^{2} e_{31}^{2} } \right)V\pi^{2} } \right) \\ \end{aligned}$$

(66)

$$\begin{aligned} S_{9} = & - \frac{3}{128}\frac{1}{{\mu_{F33} a^{3} \eta_{B33} b^{3} }}\left( {h^{4} \pi^{4} \left( {\left( {\left( {n^{4} (C_{22B} + 2C_{22F} )a^{4} + \frac{2}{9}b^{2} m^{2} n^{2} (C_{12B} } \right.} \right.} \right.} \right. \\ & \left. { + \;2C_{66B} + 2C_{12F} + 4C_{66F} )a^{2} + b^{4} m^{4} (C_{11B} + 2C_{11F} )} \right)\mu_{F33} + \frac{8}{3}a^{4} n^{4} q_{31}^{2} \\ & \left. { + \;\frac{16}{27}a^{2} b^{2} m^{2} n^{2} q_{31}^{2} + \frac{8}{3}b^{4} m^{4} q_{31}^{2} } \right)\eta_{B33} + \left( {\frac{4}{3}a^{4} n^{4} e_{31}^{2} + \frac{8}{27}a^{2} b^{2} m^{2} n^{2} e_{31}^{2} } \right. \\ & \left. {\left. {\left. { + \;\frac{4}{3}b^{4} m^{4} e_{31}^{2} } \right)\mu_{F33} } \right)W^{3} } \right) \\ \end{aligned}$$

(67)

$$\begin{aligned} S_{10} = & \frac{4351}{1377810}\frac{1}{{\mu_{F33} a^{3} \eta_{B33} b^{3} }}\left( {\left( {h^{2} \left( {\left( { - \frac{5}{34808}\mu_{F33} (2186C_{22F} + C_{22B} )\eta_{B33} } \right.} \right.} \right.} \right. \\ & \left. { + \;q_{31} \left( { - \frac{5465}{17404}q_{31} + q_{15} } \right)\eta_{F33} + \frac{23}{4351}e_{31} \left( { - \frac{5}{184}e_{31} + e_{15} } \right)\mu_{B33} } \right)n^{4} a^{4} \\ & + \;2m^{2} b^{2} \left( { - \frac{5}{34808}\mu_{F33} (C_{12B} + 2C_{66B} + 2186C_{12F} + 4372C_{66F} )\eta_{B33} } \right. \\ & \left. { + \;q_{31} \left( { - \frac{5465}{17404}q_{31} + q_{15} } \right)\eta_{F33} + \frac{23}{4351}e_{31} \left( { - \frac{5}{184}e_{31} + e_{15} } \right)\mu_{B33} } \right)n^{2} a^{2} \\ & + \;m^{4} b^{4} \left( { - \frac{5}{34808}\mu_{F33} (C_{11B} + 2186C_{11F} )\eta_{B33} + q_{31} \left( { - \frac{5465}{17404}q_{31} + q_{15} } \right)\eta_{F33} } \right. \\ & \left. {\left. { + \;\frac{23}{4351}e_{31} \left( { - \frac{5}{184}e_{31} + e_{15} } \right)\mu_{B33} } \right)} \right)\pi^{2} - \frac{106596}{4351}\left( {n^{2} \left( {C_{44B} + \frac{34}{47}C_{44F} } \right)a^{2} } \right. \\ & \left. {\left. {\left. { + \;b^{2} m^{2} \left( {C_{55B} + \frac{34}{47}C_{55F} } \right)} \right)a^{2} \mu_{F33} b^{2} \eta_{B33} } \right)h^{2} \pi^{2} W} \right) \\ \end{aligned}$$

(68)

$$\begin{aligned} S_{11} = & - \frac{2}{27}\frac{1}{{a^{2} b\mu_{F33} \eta_{B33} n}}\left( {UWm\pi^{2} h^{3} \left( {2n^{2} \left( {\left( {\left( {\frac{3}{4}C_{12B} + \frac{3}{2}C_{12F} } \right)\mu_{F33} + 2q_{31}^{2} } \right)\eta_{B33} } \right.} \right.} \right. \\ & \left. { + \;e_{31}^{2} \mu_{F33} } \right)a^{2} ( - 1)^{1 + 3n} + \left( {\left( {\left( {\frac{9}{8}n^{2} (C_{66B} + 2C_{66F} } \right)a^{2} + \frac{9}{4}b^{2} m^{2} (C_{11B} + 2C_{11F} )} \right)\mu_{F33} } \right. \\ & \left. {\left. { + \;6b^{2} m^{2} q_{31}^{2} } \right)\eta_{B33} + 3b^{2} m^{2} e_{31}^{2} \mu_{F33} } \right)( - 1)^{1 + n} + \left( {\left( {\left( {\frac{3}{8}n^{2} (C_{66B} + 2C_{66F} )a^{2} } \right.} \right.} \right. \\ & \left. {\left. {\left. { + \;\frac{3}{4}b^{2} m^{2} (C_{11B} + 2C_{11F} )} \right)\mu_{F33} + 2b^{2} m^{2} q_{31}^{2} } \right)\eta_{B33} + b^{2} m^{2} e_{31}^{2} \mu_{F33} } \right)( - 1)^{3n} \\ & + \;3n^{2} \left( {\left( {\left( {\frac{3}{4}C_{12B} + \frac{3}{2}C_{12F} } \right)\mu_{F33} + 2q_{31}^{2} } \right)\eta_{B33} + e_{31}^{2} \mu_{F33} } \right)a^{2} ( - 1)^{n} \\ & + \;\left( {\left( { - \frac{3}{4}n^{2} (C_{12B} + 2C_{12F} - 2C_{66F} - C_{66B} )a^{2} + \frac{3}{2}b^{2} m^{2} (C_{11B} + 2C_{11F} )} \right)\mu_{F33} } \right. \\ & \left. {\left. {\left. { - \;2q_{31}^{2} (a^{2} n^{2} - 2b^{2} m^{2} )} \right)\eta_{B33} - e_{31}^{2} \mu_{F33} (a^{2} n^{2} - 2b^{2} m^{2} )} \right)} \right) \\ \end{aligned}$$

(69)

$$\begin{aligned} S_{12} = & - \frac{2}{27}\frac{1}{{ab^{2} \mu_{F33} \eta_{B33} m}}\left( {VW\pi^{2} \left( {2m^{2} b^{2} \left( {\left( {\left( {\frac{3}{4}C_{12B} + \frac{3}{2}C_{12F} } \right)\mu_{F33} + 2q_{31}^{2} } \right)\eta_{B33} } \right.} \right.} \right. \\ & \left. { + \;e_{31}^{2} \mu_{F33} } \right)( - 1)^{1 + 3m} + \left( {\left( {\left( {\frac{9}{8}m^{2} (C_{66B} + 2C_{66F} )b^{2} + \frac{9}{4}a^{2} n^{2} (C_{22B} + 2C_{22F} )} \right)\mu_{F33} } \right.} \right. \\ & \left. {\left. { + \;6a^{2} n^{2} q_{31}^{2} } \right)\eta_{B33} + 3a^{2} n^{2} e_{31}^{2} \mu_{F33} } \right)( - 1)^{1 + m} + \left( {\left( {\left( {\frac{3}{8}m^{2} (C_{66B} + 2C_{66F} )b^{2} } \right.} \right.} \right. \\ & \left. {\left. {\left. { + \;\frac{3}{4}a^{2} n^{2} (C_{22B} + 2C_{22F} )} \right)\mu_{F33} + 2a^{2} n^{2} q_{31}^{2} } \right)\eta_{B33} + a^{2} n^{2} e_{31}^{2} \mu_{F33} } \right)( - 1)^{3m} \\ & + \;3m^{2} b^{2} \left( {\left( {\left( {\frac{3}{4}C_{12B} + \frac{3}{2}C_{12F} } \right)\mu_{F33} + 2q_{31}^{2} } \right)\eta_{B33} + e_{31}^{2} \mu_{F33} } \right)( - 1)^{m} \\ & + \;\left( {\left( { - \frac{3}{4}m^{2} ( - C_{66B} + 2C_{12F} - 2C_{66F} + C_{12B} )b^{2} + \frac{3}{2}a^{2} n^{2} (C_{22B} + 2C_{22F} )} \right)\mu_{F33} } \right. \\ & \left. {\left. {\left. { + \;4a^{2} n^{2} q_{31}^{2} - 2m^{2} q_{31}^{2} b^{2} } \right)\eta_{B33} + 2\left( {a^{2} n^{2} - \frac{1}{2}b^{2} m^{2} } \right)\mu_{F33} e_{31}^{2} } \right)h^{3} n} \right) \\ \end{aligned}$$

(70)

$$\begin{aligned} S_{13} = & \frac{4351}{1377810}\frac{1}{{a^{2} b\mu_{F33} \eta_{B33} }}\left( {\pi h\left( {h^{2} \left( {n^{2} \left( {\frac{23}{4351}(C_{12B} + 2C_{66B} + \frac{4351}{23}C_{12F} } \right.} \right.} \right.} \right. \\ & \left. { + \;\frac{8702}{23}C_{66F} )\mu_{F33} \eta_{B33} + q_{31} (q_{15} + q_{31} )\eta_{F33} + \frac{23}{4351}\mu_{B33} e_{31} (e_{15} + e_{31} )} \right)a^{2} \\ & + \;\left( {\frac{23}{4351}\mu_{F33} (C_{11B} + \frac{4351}{23}C_{11F} )\eta_{B33} + q_{31} (q_{15} + q_{31} )\eta_{F33} + \frac{23}{4351}\mu_{B33} e_{31} } \right. \\ & \left. {\left. {\left. {\left. {(e_{15} + e_{31} )} \right)m^{2} b^{2} } \right)\pi^{2} - \frac{106596}{4351}(C_{55B} + \frac{34}{47}C_{55F} )\mu_{F33} a^{2} b^{2} \eta_{B33} } \right)Xm} \right) \\ \end{aligned}$$

(71)

$$\begin{aligned} S_{14} = & \frac{4351}{1377810}\frac{1}{{ab^{2} \mu_{F33} \eta_{B33} }}\left( {Y\left( {h^{2} \left( {\left( {\frac{23}{4351}\left( {C_{12B} + 2C_{66B} + \frac{4351}{23}C_{12F} } \right.} \right.} \right.} \right.} \right. \\ & \left. {\left. { + \;\frac{8702}{23}C_{66F} } \right)\mu_{F33} \eta_{B33} + q_{31} (q_{15} + q_{31} )\eta_{F33} + \frac{23}{4351}\mu_{B33} e_{31} (e_{15} + e_{31} )} \right)m^{2} b^{2} \\ & + \;\left( {\frac{23}{4351}\mu_{F33} \left( {C_{22B} + \frac{4351}{23}C_{22F} } \right)\eta_{B33} + q_{31} (q_{15} + q_{31} )\eta_{F33} + \frac{23}{4351}\mu_{B33} e_{31} } \right. \\ & \left. {\left. {\left. {\left. {(e_{15} + e_{31} )} \right)n^{2} a^{2} } \right)\pi^{2} - \frac{106596}{4351}\eta_{B33} b^{2} a^{2} \mu_{F33} \left( {C_{44B} + \frac{34}{47}C_{44F} } \right)} \right)\pi nh} \right) \\ \end{aligned}$$

(72)

$$\begin{aligned} S_{15} = & - \frac{1}{12}\frac{1}{ab}\left( {\left( {\left( {(\rho_{B} + 2\rho_{F} )b^{2} + \frac{1}{183708}h^{2} n^{2} \pi^{2} (\rho_{B} + 2186\rho_{F} )} \right)a^{2} } \right.} \right. \\ & \left. {\left. {\frac{1}{183708}b^{2} h^{2} m^{2} \pi^{2} (\rho_{B} + 2186\rho_{F} )} \right)h^{2} Wtt} \right) \\ \end{aligned}$$

(73)

$$\begin{aligned} S_{16} = & - \frac{17573}{1377810}\frac{1}{{a^{2} b\eta_{B33} \mu_{F33} }}\left( {W\left( {h^{2} \left( {n^{2} \left( { - \frac{23}{17573}\mu_{F33} \left( {C_{12B} + 2C_{66B} + \frac{4351}{23}C_{12F} } \right.} \right.} \right.} \right.} \right. \\ & \left. {\left. { + \frac{8702}{23}C_{66F} } \right)\eta_{B33} + q_{31} \left( {q_{15} - \frac{4351}{17573}q_{31} } \right)\eta_{F33} + \frac{2033}{35146}(e_{15} - \frac{46}{2033}e_{31} )\mu_{B33} e_{31} } \right)a^{2} \\ & + b^{2} m^{2} \left( { - \frac{23}{17573}\mu_{F33} \left( {C_{11B} + \frac{4351}{23}C_{11F} } \right)\eta_{B33} + q_{31} \left( {q_{15} - \frac{4351}{17573}q_{31} } \right)\eta_{F33} } \right. \\ & \left. {\left. {\left. {\left. { + \frac{2033}{35146}(e_{15} - \frac{46}{2033}e_{31} )\mu_{B33} e_{31} } \right)} \right)\pi^{2} + \frac{106596}{17573}\mu_{F33} b^{2} \left( {C_{55B} + \frac{34}{47}C_{55F} } \right)\eta_{B33} a^{2} } \right)\pi \;h^{2} m} \right) \\ \end{aligned}$$

(74)

$$\begin{aligned} S_{17} = & - \frac{17573}{1377810} \frac{1}{{ab\eta_{B33} \mu_{F33} }}\left( {\left( {\left( {\left( {\frac{2033}{35146}\mu_{F33} \left( {C_{11B} + \frac{35146}{2033}C_{11F} } \right)\eta_{B33} + q_{31} (q_{15} } \right.} \right.} \right.} \right. \\ & \left. {\left. { + \;q_{31} )\eta_{F33} + \frac{2033}{35146}\mu_{B33} e_{31} (e_{15} + e_{31} )} \right)m^{2} \pi^{2} h^{2} + \frac{106596}{17573}\left( {C_{55B} + \frac{34}{47}C_{55F} } \right)\mu_{F33} a^{2} \eta_{B33} } \right)b^{2} \\ & \left. {\left. { + \;\frac{2033}{35146}\mu_{F33} n^{2} a^{2} \left( {C_{66B} + \frac{35146}{2033}C_{66F} } \right)\eta_{B33} h^{2} \pi^{2} } \right)Xh} \right) \\ \end{aligned}$$

(75)

$$\begin{aligned} S_{18} = & - \frac{2033}{2755620}\frac{1}{{\eta_{B33} \mu_{F33} }}\left( {nh^{3} Y\pi^{2} m\left( {\left( {C_{12B} + C_{66B} + \frac{35146}{2033}C_{12F} } \right.} \right.} \right. \\ & \left. {\left. { + \frac{35146}{2033}C_{66F} } \right)\mu_{F33} \eta_{B33} + \frac{35146}{2033}q_{31} (q_{15} + q_{31} )\eta_{F33} + \mu_{B33} e_{31} (e_{15} + e_{31} )} \right) \\ \end{aligned}$$

(76)

$$S_{19} = \frac{1}{1377810}h^{4} \pi Wttbm(23\rho_{B} + 4351\rho_{F} )$$

(77)

$$\begin{aligned} S_{20} = & - \frac{17573}{1377810}\frac{1}{{a\eta_{B33} \mu_{F33} b^{2} }}\left( {nh^{2} \left( {h^{2} \left( {m^{2} \left( { - \frac{23}{17573}\left( {C_{12B} + 2C_{66B} + \frac{4351}{23}C_{12F} } \right.} \right.} \right.} \right.} \right. \\ & \left. {\left. { + \;\frac{8702}{23}C_{66F} } \right)\mu_{F33} \eta_{B33} + q_{31} (q_{15} - \frac{4351}{17573}q_{31} )\eta_{F33} + \frac{2033}{35146}(e_{15} - \frac{46}{2033}e_{31} )e_{31} \mu_{B33} } \right)b^{2} \\ & + \;n^{2} \left( { - \frac{23}{17573}\mu_{F33} \left( {C_{22B} + \frac{4351}{23}C_{22F} } \right)\eta_{B33} +\, q_{31} \left( {q_{15} - \frac{4351}{17573}q_{31} } \right)\eta_{F33} + \frac{2033}{35146}(e_{15} } \right. \\ & \left. {\left. {\left. {\left. {\left. { - \;\frac{46}{2033}e_{31} } \right)e_{31} \mu_{B33} } \right)a^{2} } \right)\pi^{2} + \frac{106596}{17573}\mu_{F33} b^{2} a^{2} \eta_{B33} (C_{44B} + \frac{34}{47}C_{44F} )} \right)\pi W} \right) \\ \end{aligned}$$

(78)

$$\begin{aligned} S_{21} = & - \frac{2033}{2755620}\frac{1}{{\eta_{B33} \mu_{F33} }}\left( {X\left( {\mu_{F33} \left( {C_{12B} + C_{66B} + \frac{35146}{2033}C_{12F} } \right.} \right.} \right. \\ & \left. { + \;\frac{35146}{2033}C_{66F} } \right)\eta_{B33} + \frac{35146}{2033}q_{31} (q_{15} + q_{31} )\eta_{F33} \\ & \left. {\left. { + \;\mu_{B33} e_{31} (e_{15} + e_{31} )} \right)nh^{2} m\pi^{2} } \right) \\ \end{aligned}$$

(79)