Appendix A
The detailed expressions of \({{\varvec{\upvarepsilon}}}^{\left( n \right)}\), \({{\varvec{\upeta}}}^{\left( n \right)}\), \({{\varvec{\upgamma}}}^{\left( n \right)}\), and \({{\varvec{\upchi}}}^{\left( n \right)}\) are given as follows:
$$ \left\{ \begin{gathered} \varepsilon_{11}^{\left( n \right)} = \frac{\partial u}{{\partial x_{1} }} + \beta_{{}}^{\left( n \right)} \frac{{\partial \psi_{1}^{\left( n \right)} }}{{\partial x_{1} }},\;\varepsilon_{22}^{\left( n \right)} = \frac{\partial v}{{\partial x_{2} }} + \beta_{{}}^{\left( n \right)} \frac{{\partial \psi_{2}^{\left( n \right)} }}{{\partial x_{2} }} , \hfill \\ \varepsilon_{23}^{\left( n \right)} = \frac{1}{2}\left( {\psi_{2}^{\left( n \right)} + \frac{\partial w}{{\partial x_{2} }}} \right),\varepsilon_{31}^{\left( n \right)} = \frac{1}{2}\left( {\psi_{1}^{\left( n \right)} + \frac{\partial w}{{\partial x_{1} }}} \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \varepsilon_{12}^{\left( n \right)} = \frac{1}{2}\left[ {\frac{\partial u}{{\partial x_{2} }} + \frac{\partial v}{{\partial x_{1} }} + \beta_{{}}^{\left( n \right)} \left( {\frac{{\partial \psi_{1}^{\left( n \right)} }}{{\partial x_{2} }} + \frac{{\partial \psi_{2}^{\left( n \right)} }}{{\partial x_{1} }}} \right)} \right] , \hfill \\ \end{gathered} \right. $$
(19)
$$ \left\{ {\begin{array}{*{20}l} {\;\eta _{{111}}^{{\left( n \right)}} = \frac{2}{5}\left[ {\frac{{\partial ^{2} u}}{{\partial x_{1}^{2} }} - \frac{{\partial ^{2} v}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} - 2\frac{{\partial ^{2} u}}{{\partial x_{2}^{2} }} + \beta _{{}}^{{\left( n \right)}} \left( {\frac{{\partial ^{2} \psi _{1}^{{\left( n \right)}} }}{{\partial x_{1}^{2} }} - \frac{{\partial ^{2} \psi _{2}^{{\left( n \right)}} }}{{\partial x_{1} \partial x_{2} }} - 2\frac{{\partial ^{2} \psi _{1}^{{\left( n \right)}} }}{{\partial x_{2}^{2} }}} \right)} \right]} , \hfill \\ {\;\eta _{{222}}^{{\left( n \right)}} = \frac{2}{5}\left[ {\frac{{\partial ^{2} v}}{{\partial x_{2}^{2} }} - \frac{{\partial ^{2} u}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} - 2\frac{{\partial ^{2} v}}{{\partial x_{1}^{2} }} + \beta _{{}}^{{\left( n \right)}} \left( {\frac{{\partial ^{2} \psi _{2}^{{\left( n \right)}} }}{{\partial x_{2}^{2} }} - \frac{{\partial ^{2} \psi _{1}^{{\left( n \right)}} }}{{\partial x_{1} \partial x_{2} }} - 2\frac{{\partial ^{2} \psi _{2}^{{\left( n \right)}} }}{{\partial x_{1}^{2} }}} \right)} \right]} , \hfill \\ {\;\eta _{{333}}^{{\left( n \right)}} = - \frac{1}{5}\left( {2\frac{{\partial \psi _{1}^{{\left( n \right)}} }}{{\partial x_{1} }} + \frac{{\partial \psi _{2}^{{\left( n \right)}} }}{{\partial x_{2} }} + \frac{{\partial ^{2} w}}{{\partial x_{1}^{2} }} + \frac{{\partial ^{2} w}}{{\partial x_{2}^{2} }}} \right)} , \hfill \\ \begin{gathered} \eta _{{112}}^{{\left( n \right)}} = \frac{1}{{15}}\left[ {8\frac{{\partial ^{2} u}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} + 4\frac{{\partial ^{2} v}}{{\partial x_{1}^{2} }} - 3\frac{{\partial ^{2} v}}{{\partial x_{2}^{2} }} + \beta _{{}}^{{\left( n \right)}} \left( {8\frac{{\partial ^{2} \psi _{1}^{{\left( n \right)}} }}{{\partial x_{1} \partial x_{2} }} + 4\frac{{\partial ^{2} \psi _{2}^{{\left( n \right)}} }}{{\partial x_{1}^{2} }} - 3\frac{{\partial ^{2} \psi _{2}^{{\left( n \right)}} }}{{\partial x_{2}^{2} }}} \right)} \right] , \\ \eta _{{221}}^{{\left( n \right)}} = \frac{1}{{15}}\left[ {8\frac{{\partial ^{2} v}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} + 4\frac{{\partial ^{2} u}}{{\partial x_{2}^{2} }} - 3\frac{{\partial ^{2} u}}{{\partial x_{1}^{2} }} + \beta _{{}}^{{\left( n \right)}} \left( {8\frac{{\partial ^{2} \psi _{2}^{{\left( n \right)}} }}{{\partial x_{1} \partial x_{2} }} + 4\frac{{\partial ^{2} \psi _{1}^{{\left( n \right)}} }}{{\partial x_{2}^{2} }} - 3\frac{{\partial ^{2} \psi _{1}^{{\left( n \right)}} }}{{\partial x_{1}^{2} }}} \right)} \right] , \\ \eta _{{332}}^{{\left( n \right)}} = - \frac{1}{{15}}\left[ {2\frac{{\partial ^{2} u}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} + 3\frac{{\partial ^{2} v}}{{\partial x_{2}^{2} }} + \frac{{\partial ^{2} v}}{{\partial x_{2}^{2} }} + \beta _{{}}^{{\left( n \right)}} \left( {2\frac{{\partial ^{2} \psi _{1}^{{\left( n \right)}} }}{{\partial x_{1} \partial x_{2} }} + 3\frac{{\partial ^{2} \psi _{2}^{{\left( n \right)}} }}{{\partial x_{2}^{2} }} + \frac{{\partial ^{2} \psi _{2}^{{\left( n \right)}} }}{{\partial x_{1}^{2} }}} \right)} \right] , \\ \eta _{{331}}^{{\left( n \right)}} = - \frac{1}{{15}}\left[ {2\frac{{\partial ^{2} v}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} + 3\frac{{\partial ^{2} u}}{{\partial x_{1}^{2} }} + \frac{{\partial ^{2} u}}{{\partial x_{1}^{2} }} + \beta _{{}}^{{\left( n \right)}} \left( {2\frac{{\partial ^{2} \psi _{2}^{{\left( n \right)}} }}{{\partial x_{1} \partial x_{2} }} + 3\frac{{\partial ^{2} \psi _{1}^{{\left( n \right)}} }}{{\partial x_{1}^{2} }} + \frac{{\partial ^{2} \psi _{1}^{{\left( n \right)}} }}{{\partial x_{2}^{2} }}} \right)} \right] , \\ \eta _{{113}}^{{\left( n \right)}} = \frac{1}{{15}}\left( {8\frac{{\partial \psi _{1}^{{\left( n \right)}} }}{{\partial x_{1} }} + 4\frac{{\partial ^{2} w}}{{\partial x_{1}^{2} }} - \frac{{\partial ^{2} w}}{{\partial x_{2}^{2} }} - 2\frac{{\partial \psi _{2}^{{\left( n \right)}} }}{{\partial x_{2} }}} \right) , \\ \eta _{{223}}^{{\left( n \right)}} = \frac{1}{{15}}\left( {8\frac{{\partial \psi _{2}^{{\left( n \right)}} }}{{\partial x_{2} }} + 4\frac{{\partial ^{2} w}}{{\partial x_{2}^{2} }} - \frac{{\partial ^{2} w}}{{\partial x_{1}^{2} }} - 2\frac{{\partial \psi _{1}^{{\left( n \right)}} }}{{\partial x_{1} }}} \right) , \\ \eta _{{123}}^{{\left( n \right)}} = \eta _{{132}}^{{\left( n \right)}} = \eta _{{213}}^{{\left( n \right)}} = \eta _{{231}}^{{\left( n \right)}} = \eta _{{312}}^{{\left( n \right)}} = \eta _{{321}}^{{\left( n \right)}} = \frac{1}{3}\left( {\frac{{\partial \psi _{1}^{{\left( n \right)}} }}{{\partial x_{2} }} + \frac{{\partial ^{2} w}}{{\partial x_{1} \partial x_{2} }} + \frac{{\partial \psi _{2}^{{\left( n \right)}} }}{{\partial x_{1} }}} \right) , \\ \eta _{{212}}^{{\left( n \right)}} = \eta _{{122}}^{{\left( n \right)}} = \eta _{{221}}^{{\left( n \right)}} ,\;\eta _{{121}}^{{\left( n \right)}} = \eta _{{211}}^{{\left( n \right)}} = \eta _{{112}}^{{\left( n \right)}} ,\;\eta _{{233}}^{{\left( n \right)}} = \eta _{{332}}^{{\left( n \right)}} = \eta _{{323}}^{{\left( n \right)}},\;\eta _{{313}}^{{\left( n \right)}} = \eta _{{133}}^{{\left( n \right)}} = \eta _{{331}}^{{\left( n \right)}} , \\ \;\eta _{{223}}^{{\left( n \right)}} = \eta _{{232}}^{{\left( n \right)}} = \eta _{{322}}^{{\left( n \right)}} ,\;\eta _{{113}}^{{\left( n \right)}} = \eta _{{131}}^{{\left( n \right)}} = \eta _{{311}}^{{\left( n \right)}} \;, \\ \end{gathered} \hfill \\ \end{array} } \right. $$
(20)
$$ \left\{ \begin{gathered} \gamma_{1}^{\left( n \right)} = \frac{{\partial^{2} u}}{{\partial x_{1}^{2} }} + \frac{{\partial^{2} v}}{{\partial x_{1} \partial x_{2} }} + \beta^{\left( n \right)} \left( {\frac{{\partial^{2} \psi_{1}^{\left( n \right)} }}{{\partial x_{1}^{2} }} + \frac{{\partial^{2} \psi_{2}^{\left( n \right)} }}{{\partial x_{1} \partial x_{2} }}} \right)\;, \, \hfill \\ \gamma_{2}^{\left( n \right)} = \frac{{\partial^{2} u}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} + \frac{{\partial^{2} v}}{{\partial x_{2}^{2} }} + \beta^{\left( n \right)} \left( {\frac{{\partial^{2} \psi_{2}^{\left( n \right)} }}{{\partial x_{2}^{2} }}{ + }\frac{{\partial^{2} \psi_{1}^{\left( n \right)} }}{{\partial x_{1} \partial x_{2} }}} \right),\;\gamma_{3}^{\left( n \right)} = \frac{{\partial \psi_{1}^{\left( n \right)} }}{{\partial x_{1} }} + \frac{{\partial \psi_{2}^{\left( n \right)} }}{{\partial x_{2} }} , \, \hfill \\ \end{gathered} \right. $$
(21)
$$ \left\{ {\begin{array}{*{20}l} {\chi_{11}^{\left( n \right)} = \frac{1}{2}\left( {\frac{{\partial^{2} w}}{{\partial x_{1} \partial x_{2} }} - \frac{{\partial \psi_{2}^{\left( n \right)} }}{{\partial x_{1} }}} \right), \, \chi_{22}^{\left( n \right)} = \frac{1}{2}\left( {\frac{{\partial \psi_{1}^{\left( n \right)} }}{{\partial x_{2} }} - \frac{{\partial^{2} w}}{{\partial x_{1} \partial x_{2} }}} \right), \, \chi_{33}^{\left( n \right)} = \frac{1}{2}\left( {\frac{{\partial \psi_{2}^{\left( n \right)} }}{{\partial x_{1} }} - \frac{{\partial \psi_{1}^{\left( n \right)} }}{{\partial x_{2} }}} \right)} , \hfill \\ \begin{gathered} \chi_{23}^{\left( n \right)} = \frac{1}{4}\left[ {\frac{{\partial^{2} v}}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} - \frac{{\partial^{2} u}}{{\partial x_{2}^{2} }} + \beta^{\left( n \right)} \left( {\frac{{\partial^{2} \psi_{2}^{\left( n \right)} }}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} - \frac{{\partial^{2} \psi_{1}^{\left( n \right)} }}{{\partial x_{2}^{2} }}} \right)} \right], \, \chi_{31}^{\left( n \right)} = \frac{1}{4}\left[ {\frac{{\partial^{2} v}}{{\partial x_{1}^{2} }} - \frac{{\partial^{2} u}}{{\partial x_{1} \partial x_{2} }} + \beta^{\left( n \right)} \left( {\frac{{\partial^{2} \psi_{2}^{\left( n \right)} }}{{\partial x_{1}^{2} }} - \frac{{\partial^{2} \psi_{1}^{\left( n \right)} }}{{\partial x_{1} \partial x_{2} }}} \right)} \right] , \hfill \\ \chi_{12}^{\left( n \right)} = \frac{1}{4}\left( {\frac{{\partial^{2} w}}{{\partial x_{2}^{2} }} - \frac{{\partial^{2} w}}{{\partial x_{1}^{2} }} + \frac{{\partial \psi_{1}^{\left( n \right)} }}{{\partial x_{1} }} - \frac{{\partial \psi_{2}^{\left( n \right)} }}{{\partial x_{2} }}} \right). \hfill \\ \end{gathered} \hfill \\ \end{array} } \right. $$
(22)
Appendix B
The parameters of the boundary conditions in Eq. (12) can be shown as
$$ \begin{aligned} f_{{b_{1}^{1} }} & = \sum\limits_{{n = 1}}^{2} {\left\{ {\left( {\bar{N}_{{11}}^{{\left( n \right)}} } \right.} \right. - \bar{P}_{{1,1}}^{{\left( n \right)}} - 0.5\bar{P}_{{2,2}}^{{\left( n \right)}} - a_{{i1}} \bar{T}_{{i1,1}}^{{\left( n \right)}} - 0.5a_{{i3}} \bar{T}_{{i2,2}}^{{\left( n \right)}} + \left. {0.25\bar{M}_{{13,2}}^{{\left( n \right)}} } \right)} \tau _{1} \\ & + \left. {\left[ {\bar{N}_{{12}}^{{\left( n \right)}} - 0.5\bar{P}_{{2,1}}^{{\left( n \right)}} - 0.5a_{{i3}}^{{}} \bar{T}_{{i2,1}}^{{\left( n \right)}} - a_{{i2}} \bar{T}_{{i1,2}}^{{\left( n \right)}} + 0.25\left( {\bar{M}_{{13,1}}^{{\left( n \right)}} + 2\bar{M}_{{23,2}}^{{\left( n \right)}} } \right)} \right]\tau _{2} } \right\} , \\ \end{aligned} $$
(23)
$$ \begin{aligned} f_{{b_{2}^{1} }} & = \sum\limits_{{n = 1}}^{2} {\left\{ {\left[ {\bar{N}_{{12}}^{{\left( n \right)}} } \right.} \right.} - 0.5\bar{P}_{{1,2}}^{{\left( n \right)}} - \tilde{a}_{{i1}} \bar{T}_{{i2,1}}^{{\left( n \right)}} - 0.5\tilde{a}_{{i3}} \bar{T}_{{i1,2}}^{{\left( n \right)}} - 0.25\left. {\left( {2\bar{M}_{{13,1}}^{{\left( n \right)}} + \bar{M}_{{23,2}}^{{\left( n \right)}} } \right)} \right]\tau _{1} \\ & + \left( {\bar{N}_{{22}}^{{\left( n \right)}} } \right. - 0.5\bar{P}_{{1,1}}^{{\left( n \right)}} - \bar{P}_{{2,2}}^{{\left( n \right)}} - 0.5\tilde{a}_{{i3}} \bar{T}_{{i1,1}}^{{\left( n \right)}} - 0.5\tilde{a}_{{i2}} \bar{T}_{{i2,2}}^{{\left( n \right)}} - \left. {0.25\left. {\bar{M}_{{23,1}}^{{\left( n \right)}} } \right)\tau _{2} } \right\} , \\ \end{aligned} $$
(24)
$$ \begin{aligned} f_{{b_{3}^{1} }} & = \sum\limits_{{n = 1}}^{2} {\left\{ {\left[ {\bar{N}_{{13}}^{{\left( n \right)}} } \right.} \right.} - \tilde{a}_{{i1}} \bar{T}_{{i3}}^{{\left( n \right)}} - \bar{T}_{{14,2}}^{{\left( n \right)}} - 0.25\left. {\left( {\bar{M}_{{11,2}}^{{\left( n \right)}} - 2\bar{M}_{{12,1}}^{{\left( n \right)}} - \bar{M}_{{22,2}}^{{\left( n \right)}} } \right)} \right]\tau _{1} \\ & + \quad \left[ {\bar{N}_{{23}}^{{\left( n \right)}} } \right. + a_{{i2}} \bar{T}_{{i3,2}}^{{\left( n \right)}} - \bar{T}_{{14,1}}^{{\left( n \right)}} - 0.25\left. {\left. {\left( {\bar{M}_{{22,1}}^{{\left( n \right)}} + \bar{M}_{{11,1}}^{{\left( n \right)}} + 2\bar{M}_{{21,2}}^{{\left( n \right)}} } \right)} \right]\tau _{2} } \right\} , \\ \end{aligned} $$
(25)
$$ f_{{b_{4}^{1} }}^{{}} = f_{{b_{4}^{1} }}^{\left( 1 \right)} ,\;f_{{b_{5}^{1} }}^{{}} = f_{{b_{5}^{1} }}^{\left( 1 \right)} ,f_{{b_{6}^{1} }}^{{}} = f_{{b_{4}^{1} }}^{\left( 2 \right)} ,\;f_{{b_{7}^{1} }}^{{}} = f_{{b_{5}^{1} }}^{\left( 2 \right)} ,\;f_{{b_{8}^{1} }}^{{}} = f_{{b_{8}^{1} }}^{\left( 1 \right)} ,\;f_{{b_{9}^{1} }}^{{}} = f_{{b_{8}^{1} }}^{\left( 2 \right)} , $$
(26)
$$ \begin{gathered} f_{{b_{4}^{1} }}^{\left( n \right)} = \left[ {N_{11}^{\left( n \right)} - b_{{\tilde{i}1}} P_{{\tilde{i},1}}^{\left( n \right)} + \overline{P}_{3}^{{}} - a_{i1} T_{i1,1}^{\left( n \right)} - 0.5a_{i3} T_{i2,2}^{\left( n \right)} - a_{i4} \overline{T}_{i3}^{{}} + 0.25\left( {\overline{M}_{13,2}^{\left( n \right)} + 2\overline{M}_{12}^{\left( n \right)} } \right)} \right]\tau_{1} \\ +\left[ {N_{12}^{\left( n \right)} - 0.5P_{2,1}^{\left( n \right)} - 0.5a_{i3} T_{i2,1}^{\left( n \right)} - a_{i2} T_{i1,2}^{\left( n \right)} + {\kern 1pt} {\kern 1pt} 2\overline{T}_{14}^{\left( n \right)} + 0.25\left( {M_{13,1}^{\left( n \right)} + 2M_{23,2}^{\left( n \right)} + 2\overline{M}_{22}^{\left( n \right)} - 2\overline{M}_{33}^{\left( n \right)} } \right)} \right]\tau_{2} , \\ \end{gathered} $$
(27)
$$ \begin{gathered} f_{{b_{5}^{1} }}^{\left( n \right)} = \left[ {N_{12}^{\left( n \right)} } \right. - 0.5P_{1,2}^{\left( n \right)} - 0.5\tilde{a}_{i3} T_{i1,2}^{\left( n \right)} - \tilde{a}_{i1} T_{i2,1}^{\left( n \right)} {\kern 1pt} + 2\overline{T}_{14}^{\left( n \right)} - 0.25\left. {\left( {M_{23,2}^{\left( n \right)} + {\kern 1pt} 2M_{13,1}^{\left( n \right)} + 2\overline{M}_{11}^{\left( n \right)} - 2\overline{M}_{33}^{\left( n \right)} } \right)} \right] \\ {\kern 1pt} \tau_{1} {\kern 1pt} + {\kern 1pt} \left[ {N_{22}^{\left( n \right)} } \right. + \overline{P}_{3}^{\left( n \right)} - 0.5P_{1,1}^{\left( n \right)} - P_{2,2}^{\left( n \right)} - 0.5\tilde{a}_{i3} T_{i1,1}^{\left( n \right)} - \tilde{a}_{i2} T_{i2,2}^{\left( n \right)} - \tilde{a}_{i4} \overline{T}_{i3}^{\left( n \right)} - \left. {0.25\left( {2\overline{M}_{12}^{\left( n \right)} + M_{23,1}^{\left( n \right)} } \right)} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} \tau_{2} , \\ \end{gathered} $$
(28)
$$ f_{{b_{8}^{1} }}^{\left( n \right)} = Q_{1}^{\left( n \right)} \tau_{1} + Q_{2}^{\left( n \right)} \tau_{2} , $$
(29)
$$ \, f_{{b_{1}^{2} }} = \sum\limits_{n = 1}^{2} {\left[ {\left( {\left. {a_{i1} \overline{T}_{i1}^{\left( n \right)} + \overline{P}_{1}^{\left( n \right)} } \right)\tau_{1} + 0.25\left( {2\overline{P}_{2}^{\left( n \right)} + 2a_{i3} \overline{T}_{i2}^{\left( n \right)} - \left. {\overline{M}_{13}^{\left( n \right)} } \right)} \right.\tau_{2} } \right.} \right]} , $$
(30)
$$ f_{{b_{2}^{2} }} = 0.25\sum\limits_{n = 1}^{2} {\left[ {\left( {{\kern 1pt} 4\tilde{a}_{i1} \overline{T}_{i2}^{\left( n \right)} + 2\overline{M}_{13}^{\left( n \right)} } \right)\tau_{1} + \left( {2\overline{P}_{1}^{\left( n \right)} } \right. + 2\tilde{a}_{i3} \overline{T}_{i1}^{\left( n \right)} + \left. {\overline{M}_{23}^{\left( n \right)} } \right)\tau_{2} } \right]} , $$
(31)
$$ f_{{b_{3}^{2} }} = {\kern 1pt} 0.25\sum\limits_{n = 1}^{2} {\left[ {\left( {4\tilde{a}_{i1} \overline{T}_{i3}^{\left( n \right)} } \right. - 2\left. {\overline{M}_{12}^{\left( n \right)} } \right)\tau_{1} + {\kern 1pt} \left( {2\overline{T}_{14}^{\left( n \right)} + \overline{M}_{11}^{\left( n \right)} - \overline{M}_{22}^{\left( n \right)} } \right)\tau_{2} } \right]} , $$
(32)
$$ f_{{b_{4}^{2} }}^{{}} = f_{{b_{4}^{2} }}^{\left( 1 \right)} ,\;f_{{b_{5}^{2} }}^{{}} = f_{{b_{5}^{2} }}^{\left( 1 \right)} ,\;f_{{b_{6}^{2} }}^{{}} = f_{{b_{4}^{2} }}^{\left( 2 \right)} ,\;f_{{b_{7}^{2} }}^{{}} = f_{{b_{5}^{2} }}^{\left( 2 \right)} , $$
(33)
$$ f_{{b_{4}^{2} }}^{\left( n \right)} = \left( {P_{1}^{\left( n \right)} {\kern 1pt} + a_{i1} T_{i1}^{\left( n \right)} } \right)\tau_{1} + 0.25\left( {{\kern 1pt} 2P_{2}^{\left( n \right)} } \right. + 2a_{i3} T_{i2}^{\left( n \right)} - {\kern 1pt} \left. {M_{13}^{\left( n \right)} } \right)\tau_{2}, $$
(34)
$$ f_{{b_{5}^{2} }}^{\left( n \right)} = \left( {\tilde{a}_{i1} T_{i2}^{\left( n \right)} + 0.5M_{13}^{\left( n \right)} {\kern 1pt} } \right)\tau_{1} {\kern 1pt} + {\kern 1pt} {\kern 1pt} 0.25\left( {2P_{1}^{\left( n \right)} } \right. + 2\tilde{a}_{i3} T_{i1}^{\left( n \right)} + \left. {M_{23}^{\left( n \right)} {\kern 1pt} } \right)\tau_{2} , $$
(35)
$$ f_{{b_{1}^{3} }} = 0.25\sum\limits_{n = 1}^{2} {\left[ {\left( {2\overline{P}_{2}^{\left( n \right)} } \right. + 2a_{i3}^{{}} \overline{T}_{i2}^{\left( n \right)} - \left. {\overline{M}_{13}^{\left( n \right)} } \right)\tau_{1} + {\kern 1pt} {\kern 1pt} 2\left( {a_{i2} \overline{T}_{i1}^{\left( n \right)} \left. { - \overline{M}_{23}^{\left( n \right)} } \right)\tau_{2} } \right.} \right]} , $$
(36)
$$ f_{{b_{2}^{3} }} = 0.25\sum\limits_{n = 1}^{2} {\left[ {\left( {2\overline{P}_{1}^{\left( n \right)} - 2\tilde{a}_{i3} \overline{T}_{i1}^{\left( n \right)} + \overline{M}_{23}^{\left( n \right)} } \right)\tau_{1} + {\kern 1pt} 2\left( {2\overline{P}_{2}^{\left( n \right)} + \tilde{a}_{i2} \overline{T}_{i2}^{\left( n \right)} } \right)\tau_{2} } \right]} , $$
(37)
$$ f_{{b_{3}^{3} }} = 0.25\sum\limits_{n = 1}^{2} {\left[ {\left( {4\overline{T}_{14}^{\left( n \right)} + {\kern 1pt} \overline{M}_{11}^{\left( n \right)} - \overline{M}_{22}^{\left( n \right)} } \right)\tau_{1} + 2\left( {{\kern 1pt} 2a_{i2} \overline{T}_{i3}^{\left( n \right)} + \overline{M}_{12}^{\left( n \right)} } \right)\tau_{2} } \right]} , $$
(38)
$$ f_{{b_{4}^{3} }}^{{}} = f_{{b_{4}^{3} }}^{\left( 1 \right)} ,\;f_{{b_{5}^{3} }}^{{}} = f_{{b_{5}^{3} }}^{\left( 1 \right)} ,\;f_{{b_{6}^{3} }}^{{}} = f_{{b_{4}^{3} }}^{\left( 2 \right)} ,\;f_{{b_{7}^{3} }}^{{}} = f_{{b_{5}^{3} }}^{\left( 2 \right)} , $$
(39)
$$ f_{{b_{4}^{3} }}^{\left( n \right)} = 0.25\left( {2P_{2}^{\left( n \right)} } \right. + 2a_{i3} T_{i2}^{\left( n \right)} - \left. {M_{13}^{\left( n \right)} } \right)\tau_{1} + {\kern 1pt} {\kern 1pt} \left( {a_{i2} T_{i1}^{\left( n \right)} - 0.5M_{23}^{\left( n \right)} {\kern 1pt} } \right)\tau_{2} , $$
(40)
$$ f_{{b_{5}^{3} }}^{\left( n \right)} = 0.25\left( {2P_{1}^{\left( n \right)} + 2\tilde{a}_{i3} T_{i1}^{\left( n \right)} - M_{23}^{\left( n \right)} } \right)\tau_{1} {\kern 1pt} + {\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {P_{2}^{\left( n \right)} + 0.5\tilde{a}_{i2} T_{i2}^{\left( n \right)} } \right)\tau_{2} , $$
(41)
where \(\tau_{i} \;\left( {i = 1,2} \right)\) denote the direction cosines of the unit vector normal to the boundary of the middle plane.
Appendix C
In Eq. (15), the partial differential operator \({\mathbb{Z}}_{IJ}\) can be written as
$$ {\mathbb{Z}}_{IJ} = \sum\limits_{{\hat{n} = 0}}^{4} {\sum\limits_{{\hat{k} = 0}}^{{4 - \hat{n}}} {\left( {{\text{A}}_{{IJ\hat{n}\hat{k}}} \frac{{\partial^{{\hat{n} + \hat{k}}} }}{{\partial \overline{x}_{1}^{{\hat{n}}} \partial \overline{x}_{2}^{{\hat{k}}} }}} \right)} } ;\quad \left( {0 \le \hat{n} + \hat{k} \le 4;\;I,\;J = 1,\;2,\; \cdots ,9} \right) $$
(42)
where \(\hat{n},\;\hat{k}\) denote the derivation order to \(\overline{x}_{1} ,\;\overline{x}_{2}\). In our model, the nonzero coefficients \({\text{A}}_{{IJ\hat{n}\hat{k}}}\) can be expressed as
$$ \begin{gathered} {\text{A}}_{1120} = - \sum\limits_{n = 1}^{2} {\overline{c}_{11}^{\left( n \right)} \theta^{\left( n \right)} } \lambda ,\;{\text{A}}_{1140} = \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{1140} } {\mathbf{l}}^{\left( n \right)} \theta^{\left( n \right)} \lambda^{3} ,{\text{A}}_{1104} = \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{1104} } {\mathbf{l}}^{\left( n \right)} \theta^{\left( n \right)} \lambda^{3} \eta^{4}, \hfill \\ {\text{A}}_{1102} = - \sum\limits_{n = 1}^{2} {\overline{c}_{66}^{\left( n \right)} \theta^{\left( n \right)} } \lambda \eta^{2} ,\;{\text{A}}_{1122} = \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{1122} } {\mathbf{l}}^{\left( n \right)} \theta^{\left( n \right)} \lambda^{3} \eta^{2} ,{\text{A}}_{1211} = - \sum\limits_{n = 1}^{2} {\left( {\overline{c}_{12}^{\left( n \right)} \theta^{\left( n \right)} + \overline{c}_{66}^{\left( n \right)} \theta^{\left( n \right)} } \right)\lambda \eta } , \hfill \\ \end{gathered} $$
$$ {\text{A}}_{1213} = \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{1213} {\mathbf{l}}^{\left( n \right)} \theta_{{}}^{\left( n \right)} } \lambda^{3} \eta^{3} ,\;{\text{A}}_{1231} = \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{1213} {\mathbf{l}}^{\left( n \right)} \theta_{{}}^{\left( n \right)} } \lambda^{3} \eta ,{\text{A}}_{1420} = - 0.5\overline{c}_{11}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{2} , $$
$$ \begin{gathered} {\text{A}}_{1402} = - 0.5\overline{c}_{66}^{\left( 1 \right)} \lambda^{2} \eta^{2} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} ,\;{\text{A}}_{1440} = {\mathbf{a}}_{1440} {\mathbf{l}}^{\left( 1 \right)} \lambda^{4} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \;{\text{A}}_{1404} = {\mathbf{a}}_{1404} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{4} \eta^{4}, \hfill \\ {\text{A}}_{1422} = {\mathbf{a}}_{1422} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{4} \eta^{2} ,{\text{A}}_{1511} = - 0.5\left( {\overline{c}_{12}^{\left( 1 \right)} + \overline{c}_{66}^{\left( 1 \right)} } \right)\left( {\theta^{\left( 1 \right)} } \right)^{2} \lambda^{2} \eta ,\;{\text{A}}_{1513} = {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta^{\left( 1 \right)} } \right)^{2} \lambda^{4} \eta^{3} , \hfill \\ \end{gathered} $$
$$ \begin{gathered} {\text{A}}_{1531} = {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta^{\left( 1 \right)} } \right)^{2} \lambda^{4} \eta ,{\text{A}}_{1620} = 0.5\overline{c}_{11}^{\left( 2 \right)} \left( {\theta^{\left( 2 \right)} } \right)^{2} \lambda^{2} ,{\text{A}}_{1602} = 0.5\overline{c}_{66}^{\left( 2 \right)} \left( {\theta^{\left( 2 \right)} } \right)^{2} \lambda^{2} \eta^{2}, \hfill \\ {\text{A}}_{1640} = {\mathbf{a}}_{1640} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta^{\left( 2 \right)} } \right)^{2} \lambda^{4} ,\;{\text{A}}_{1604} = {\mathbf{a}}_{1604} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta^{\left( 2 \right)} } \right)^{2} \lambda^{4} \eta^{4} ,\;{\text{A}}_{1622} = - {\mathbf{a}}_{1422} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta^{\left( 2 \right)} } \right)^{2} \lambda^{4} \eta^{2} , \hfill \\ \end{gathered} $$
$$ {\text{A}}_{1711} = 0.5\left( {\overline{c}_{12}^{\left( 2 \right)} + \overline{c}_{66}^{\left( 2 \right)} } \right)\left( {\theta^{\left( 2 \right)} } \right)^{2} \lambda^{2} \eta ,\;{\text{A}}_{1713} = - {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta^{\left( 2 \right)} } \right)^{2} \lambda^{4} \eta^{3} ,\;{\text{A}}_{1731} = - {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta^{\left( 2 \right)} } \right)^{2} \lambda^{4} \eta , $$
$$ {\text{A}}_{2111} = - \sum\limits_{n = 1}^{2} {\left( {\overline{c}_{12}^{\left( n \right)} + \overline{c}_{66}^{\left( n \right)} } \right)\theta_{{}}^{\left( n \right)} \lambda \eta } ,\;{\text{A}}_{2131} = \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{1213} } {\mathbf{l}}^{\left( n \right)} \theta_{{}}^{\left( n \right)} \lambda^{3} \eta ,\;{\text{A}}_{2113} = \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{1213} } {\mathbf{l}}^{\left( n \right)} \theta_{{}}^{\left( n \right)} \lambda^{3} \eta^{3} , $$
$$ \begin{gathered} {\text{A}}_{2202} = - \sum\limits_{n = 1}^{2} {\overline{c}_{22}^{\left( n \right)} \theta_{{}}^{\left( n \right)} } \lambda \eta^{2} ,\;{\text{A}}_{2201} = - \sum\limits_{n = 1}^{2} {\overline{c}_{66}^{\left( n \right)} \theta_{{}}^{\left( n \right)} } \lambda ,\;{\text{A}}_{2204} = \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{1140} } {\mathbf{l}}^{\left( n \right)} \theta_{{}}^{\left( n \right)} \lambda^{3} \eta^{4} , \hfill \\ {\text{A}}_{2240} = \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{2240} {\mathbf{l}}^{\left( n \right)} \theta_{{}}^{\left( n \right)} } \lambda^{3} ,\;{\text{A}}_{2222} = \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{1122} } {\mathbf{l}}^{\left( n \right)} \theta_{{}}^{\left( n \right)} \lambda^{3} \eta^{2} ,\;{\text{A}}_{2411} = - 0.5\left( {\overline{c}_{12}^{\left( 1 \right)} + \overline{c}_{66}^{\left( 1 \right)} } \right)\left( {\theta_{{}}^{\left( n \right)} } \right)^{2} \lambda^{2} \eta \; , \hfill \\ \end{gathered} $$
$$ {\text{A}}_{2431} = {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{4} \eta ,\;{\text{A}}_{2413} = {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{4} \eta^{3} ,\;{\text{A}}_{2502} = - 0.5\overline{c}_{22}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{2} \eta^{2} , $$
$$ {\text{A}}_{2520} = - 0.5\overline{c}_{66}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{2} ,\;{\text{A}}_{2504} = {\mathbf{a}}_{1440} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{4} \eta^{4} ,\;{\text{A}}_{2540} = {\mathbf{a}}_{1404} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{4} , $$
$$ {\text{A}}_{2522} = {\mathbf{a}}_{1422} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{4} \eta^{2} ,{\text{A}}_{2611} = 0.5\left( {\overline{c}_{12}^{\left( 2 \right)} + \overline{c}_{66}^{\left( 2 \right)} } \right)\left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{2} \eta ,\;{\text{A}}_{2631} = - {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{4} \eta , $$
$$ {\text{A}}_{2613} = - {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{4} \eta^{3} ,\;{\text{A}}_{2702} = 0.5\overline{c}_{22}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{2} \eta^{2} ,{\text{A}}_{2720} = 0.5\overline{c}_{66}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{2} , $$
$$ {\text{A}}_{2704} = {\mathbf{a}}_{1640} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{4} \eta^{4} ,{\text{A}}_{2740} = {\mathbf{a}}_{1604} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{4} ,\;{\text{A}}_{2722} = - {\mathbf{a}}_{1422} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{4} \eta^{2} , $$
$$ \begin{gathered} {\text{A}}_{3320} = \left[ {\sum\limits_{n = 1}^{2} {\overline{c}_{44}^{\left( n \right)} \theta_{{}}^{\left( n \right)} } - \left( {\overline{k}_{g} + \overline{\Gamma }} \right)} \right]\lambda ,\;{\text{A}}_{3340} = - \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{1104} } {\mathbf{l}}^{\left( n \right)} \theta_{{}}^{\left( n \right)} \lambda^{3} ,\;{\text{A}}_{3302} = \left[ {\sum\limits_{n = 1}^{2} {\overline{c}_{44}^{\left( n \right)} \theta_{{}}^{\left( n \right)} } - \left( {\overline{k}_{g} + \overline{\Gamma }} \right)} \right]\lambda \eta^{2}, \hfill \\ {\text{A}}_{3304} = - \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{1104} } {\mathbf{l}}^{\left( n \right)} \theta_{{}}^{\left( n \right)} \lambda^{3} \eta^{4} ,\;{\text{A}}_{3322} = \sum\limits_{n = 1}^{2} {{\mathbf{a}}_{3322} } {\mathbf{l}}^{\left( n \right)} \theta_{{}}^{\left( n \right)} \lambda^{3} \eta^{2} ,{\text{A}}_{3410} = \overline{c}_{44}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} \lambda ,\;{\text{A}}_{3430} = {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} \lambda^{3} , \hfill \\ \end{gathered} $$
$$ A_{3412} = {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} \lambda^{3} \eta^{2} ,\;{\text{A}}_{3501} = \overline{c}_{44}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} \lambda \eta ,\;{\text{A}}_{3521} = {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} \lambda^{3} \eta ,\;A_{3503} = {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} \lambda^{3} \eta^{3} , $$
$$ {\text{A}}_{3610} = \overline{c}_{44}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} \lambda ,\;{\text{A}}_{3630} = {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} \lambda^{3} ,\;A_{3612} = {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} \lambda^{3} \eta^{2} {\text{A}}_{3701} = \overline{c}_{44}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} \lambda \eta , $$
$$ {\text{A}}_{3721} = {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} \lambda^{3} \eta A_{3702} = {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} \lambda^{3} \eta^{3} {\text{A}}_{3820} = \left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\theta_{{}}^{\left( 1 \right)} \overline{e}_{15} \lambda^{2} ,\;{\text{A}}_{3802} = \left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\theta_{{}}^{\left( 1 \right)} \overline{e}_{24} \lambda^{2} \eta^{2} , $$
$$ {\text{A}}_{3920} = \left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\theta_{{}}^{\left( 2 \right)} \overline{f}_{15} \lambda^{2} ,\;{\text{A}}_{3902} = \left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\theta_{{}}^{\left( 2 \right)} \overline{f}_{24} \lambda^{2} \eta^{2} {\text{A}}_{4120} = - 0.5\overline{c}_{11}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda , $$
$$ {\text{A}}_{4102} = - 0.5\overline{c}_{66}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda \eta^{2} ,\;{\text{A}}_{4140} = {\mathbf{a}}_{1440} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{3} ,{\text{A}}_{4104} = {\mathbf{a}}_{1404} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{3} \eta^{4} , $$
$$ {\text{A}}_{4122} = {\mathbf{a}}_{1422} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{3} \eta^{2} ,{\text{A}}_{4211} = - 0.5\left( {\overline{c}_{12}^{\left( 1 \right)} + \overline{c}_{66}^{\left( 1 \right)} } \right)\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda \eta ,\;{\text{A}}_{4213} = {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{3} \eta^{3} , $$
$$ {\text{A}}_{4231} = {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{3} \eta ,{\text{A}}_{4310} = \overline{c}_{44}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} ,\;{\text{A}}_{4330} = - {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} \lambda^{2} ,\;{\text{A}}_{4312} = - {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} \lambda^{2} \eta^{2} , $$
$$ \begin{gathered} {\text{A}}_{4400} = \overline{c}_{44}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} ,{\text{A}}_{4420} = - \left[ {{{\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \overline{c}_{11}^{\left( 1 \right)} } \mathord{\left/ {\vphantom {{\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \overline{c}_{11}^{\left( 1 \right)} } 3}} \right. \kern-\nulldelimiterspace} 3} + {\mathbf{a}}_{4420} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} } \right]\lambda^{2} ,\;\;{\text{A}}_{4422} = {\mathbf{a}}_{4422} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \lambda^{4} \eta^{2}, \hfill \\ {\text{A}}_{4402} = - \left[ {{{\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \overline{c}_{66}^{\left( 1 \right)} } \mathord{\left/ {\vphantom {{\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \overline{c}_{66}^{\left( 1 \right)} } 3}} \right. \kern-\nulldelimiterspace} 3} + {\mathbf{a}}_{4402} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} } \right]\lambda^{2} \eta^{2} ,\;{\text{A}}_{4440} = {\mathbf{a}}_{4440} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \lambda^{4} ,\;{\text{A}}_{4404} = {\mathbf{a}}_{4404} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \lambda^{4} \eta^{4} , \hfill \\ \end{gathered} $$
$$ {\text{A}}_{4511} = \left[ { - {{\left( {\overline{c}_{12}^{\left( 1 \right)} + \overline{c}_{66}^{\left( 1 \right)} } \right)\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} } \mathord{\left/ {\vphantom {{\left( {\overline{c}_{12}^{\left( 1 \right)} + \overline{c}_{66}^{\left( 1 \right)} } \right)\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} } 3}} \right. \kern-\nulldelimiterspace} 3} + {\mathbf{a}}_{4511} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} } \right]\lambda^{2} \eta ,\;{\text{A}}_{4531} = {\mathbf{a}}_{4531} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \lambda^{4} \eta^{3} , $$
$$ \begin{gathered} {\text{A}}_{4513} = {\mathbf{a}}_{4531} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \lambda^{4} \eta ,\;{\text{A}}_{4810} = - \left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\left( {\overline{e}_{15} + \overline{e}_{31} } \right)\theta_{{}}^{\left( 1 \right)} \lambda \;{\text{A}}_{5111} = - 0.5\left( {\overline{c}_{12}^{\left( 1 \right)} + \overline{c}_{66}^{\left( 1 \right)} } \right)\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda \eta, \hfill \\ {\text{A}}_{5113} = {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{3} \eta^{3} {\text{, A}}_{5131} = {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{3} \eta ,\;{\text{A}}_{5202} = - 0.5\overline{c}_{22}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda \eta^{2} , \hfill \\ \end{gathered} $$
$$ {\text{A}}_{5220} = - 0.5\overline{c}_{66}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda ,\;{\text{A}}_{5222} = {\mathbf{a}}_{1422} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{3} \eta^{4} ,{\text{A}}_{5204} = {\mathbf{a}}_{1440} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{3} , $$
$$ {\text{A}}_{5240} = {\mathbf{a}}_{1404} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{2} \lambda^{3} \eta^{2} ,\;{\text{A}}_{5301} = \overline{c}_{44}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} \eta ,\;{\text{A}}_{5303} = - {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} \lambda^{2} \eta^{3} ,\;{\text{A}}_{5321} = - {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} \lambda^{2} \eta , $$
$$ {\text{A}}_{5411} = \left[ {{{ - \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \left( {\overline{c}_{12}^{\left( 1 \right)} + \overline{c}_{66}^{\left( 1 \right)} } \right)} \mathord{\left/ {\vphantom {{ - \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \left( {\overline{c}_{12}^{\left( 1 \right)} + \overline{c}_{66}^{\left( 1 \right)} } \right)} 3}} \right. \kern-\nulldelimiterspace} 3} + {\mathbf{a}}_{4511} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} } \right]\lambda^{2} \eta ,\;{\text{A}}_{5431} = {\mathbf{a}}_{4531} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \lambda^{4} \eta , $$
$$ \begin{gathered} {\text{A}}_{5413} = {\mathbf{a}}_{4531} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \lambda^{4} \eta^{3} ,\;{\text{A}}_{5500} = \overline{c}_{44}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} ,\;{\text{A}}_{5502} = \left[ { - {{\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \overline{c}_{22}^{\left( 1 \right)} } \mathord{\left/ {\vphantom {{\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \overline{c}_{22}^{\left( 1 \right)} } 3}} \right. \kern-\nulldelimiterspace} 3} + {\mathbf{a}}_{4420} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} } \right]\lambda^{2} \eta^{2}, \hfill \\ {\text{A}}_{5520} = \left[ { - {{\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \overline{c}_{66}^{\left( 1 \right)} } \mathord{\left/ {\vphantom {{\left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \overline{c}_{66}^{\left( 1 \right)} } 3}} \right. \kern-\nulldelimiterspace} 3} + {\mathbf{a}}_{4402} {\mathbf{l}}^{\left( 1 \right)} \theta_{{}}^{\left( 1 \right)} } \right]\lambda^{2} ,{\text{A}}_{5522} = {\mathbf{a}}_{4422} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \lambda^{4} \eta^{2} ,\;{\text{A}}_{5504} = {\mathbf{a}}_{4440} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \lambda^{4} \eta^{4}, \hfill \\ {\text{A}}_{5540} = {\mathbf{a}}_{4404} {\mathbf{l}}^{\left( 1 \right)} \left( {\theta_{{}}^{\left( 1 \right)} } \right)^{3} \lambda^{4} ,\;{\text{A}}_{5801} = - \left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\left( {\overline{e}_{24} + \overline{e}_{31} } \right)\theta_{{}}^{\left( 1 \right)} \lambda \eta ,{\text{A}}_{6120} = 0.5\overline{c}_{11}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda , \hfill \\ \end{gathered} $$
$$ \begin{gathered} {\text{A}}_{6102} = 0.5\overline{c}_{66}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda \eta^{2} ,\;{\text{A}}_{6140} = {\mathbf{a}}_{1440} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{3} ,{\text{A}}_{6104} = {\mathbf{a}}_{1404} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{3} \eta^{4}, \hfill \\ {\text{A}}_{6122} = {\mathbf{a}}_{1422} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{3} \eta^{2} ,{\text{A}}_{6211} = 0.5\left( {\overline{c}_{12}^{\left( 2 \right)} + \overline{c}_{66}^{\left( 2 \right)} } \right)\left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda \eta ,\;{\text{A}}_{6213} = - {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{3} \eta^{3} , \hfill \\ \end{gathered} $$
$$ {\text{A}}_{6231} = - {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{3} \eta ,\;{\text{A}}_{6310} = \overline{c}_{44}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} ,\;{\text{A}}_{6330} = - {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} \lambda^{2} ,\;{\text{A}}_{6312} = - {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} \lambda^{2} \eta^{2} , $$
$$ \begin{gathered} {\text{A}}_{6600} = \overline{c}_{44}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} ,\;{\text{A}}_{6620} = \left[ {{{ - \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \overline{c}_{11}^{\left( 2 \right)} } \mathord{\left/ {\vphantom {{ - \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \overline{c}_{11}^{\left( 2 \right)} } 3}} \right. \kern-\nulldelimiterspace} 3}{ + }{\mathbf{a}}_{4420} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} } \right]\lambda^{2} ,\;{\text{A}}_{6640} = {\mathbf{a}}_{4440} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \lambda^{4}, \hfill \\ {\text{A}}_{6602} = \left[ { - {{\left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \overline{c}_{66}^{\left( 2 \right)} } \mathord{\left/ {\vphantom {{\left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \overline{c}_{66}^{\left( 2 \right)} } 3}} \right. \kern-\nulldelimiterspace} 3} + {\mathbf{a}}_{4402} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} } \right]\lambda^{2} \eta^{2} ,{\text{A}}_{6622} = {\mathbf{a}}_{4422} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \lambda^{4} \eta^{2} , \hfill \\ \end{gathered} $$
$$ \begin{gathered} {\text{A}}_{6604} = {\mathbf{a}}_{4404} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \lambda^{4} \eta^{4} {\text{A}}_{6910} = - \left( {\overline{f}_{15} + \overline{f}_{31} } \right)\left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\theta_{{}}^{\left( 2 \right)} \lambda ,{\text{A}}_{7111} = 0.5\left( {\overline{c}_{12}^{\left( 2 \right)} + \overline{c}_{66}^{\left( 2 \right)} } \right)\left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda \eta, \hfill \\ {\text{A}}_{7113} = - {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 2 \right)} \delta^{2} \lambda^{3} \eta^{3} ,\;{\text{A}}_{7131} = - {\mathbf{a}}_{1513} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{3} \eta ,\;{\text{A}}_{7202} = 0.5\overline{c}_{22}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda \eta^{2}, \hfill \\ \end{gathered} $$
$$ {\text{A}}_{7220} = 0.5\overline{c}_{66}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda ,\;{\text{A}}_{7222} = {\mathbf{a}}_{1422} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{3} \eta^{4} ,{\text{A}}_{7204} = - {\mathbf{a}}_{1440} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{3} , $$
$$ {\text{A}}_{7240} = - {\mathbf{a}}_{1404} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{2} \lambda^{3} \eta^{2} ,{\text{A}}_{7301} = \overline{c}_{44}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} \eta ,\;{\text{A}}_{7303} = - {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} \lambda^{2} \eta^{3} ,\;{\text{A}}_{7321} = - {\mathbf{a}}_{3430} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} \lambda^{2} \eta , $$
$$ {\text{A}}_{7611} = \left[ { - {{\left( {\overline{c}_{12}^{\left( 2 \right)} + \overline{c}_{66}^{\left( 2 \right)} } \right)\left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} } \mathord{\left/ {\vphantom {{\left( {\overline{c}_{12}^{\left( 2 \right)} + \overline{c}_{66}^{\left( 2 \right)} } \right)\left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} } 3}} \right. \kern-\nulldelimiterspace} 3}{ + }{\mathbf{a}}_{4511} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} } \right]\lambda^{2} \eta ,\;{\text{A}}_{7631} = {\mathbf{a}}_{4531} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \lambda^{4} \eta , $$
$$ \begin{gathered} {\text{A}}_{7613} = {\mathbf{a}}_{4531} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \lambda^{4} \eta^{3} ,\;{\text{A}}_{7700} = \overline{c}_{44}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} ,\;{\text{A}}_{7702} = \left[ { - \overline{c}_{22}^{\left( 2 \right)} {{\left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} } \mathord{\left/ {\vphantom {{\left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} } 3}} \right. \kern-\nulldelimiterspace} 3} + {\mathbf{a}}_{4420} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} } \right]\lambda^{2} \eta^{2}, \hfill \\ {\text{A}}_{7720} = \left[ { - {{\left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \overline{c}_{66}^{\left( 2 \right)} } \mathord{\left/ {\vphantom {{\left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \overline{c}_{66}^{\left( 2 \right)} } 3}} \right. \kern-\nulldelimiterspace} 3} + {\mathbf{a}}_{4402} {\mathbf{l}}^{\left( 2 \right)} \theta_{{}}^{\left( 2 \right)} } \right]\lambda^{2} ,\;{\text{A}}_{7704} = {\mathbf{a}}_{4440} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \lambda^{4} \eta^{4} ,\;{\text{A}}_{7722} = {\mathbf{a}}_{4422} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \lambda^{4} \eta^{2} , \hfill \\ \end{gathered} $$
$$ {\text{A}}_{7740} = {\mathbf{a}}_{4404} {\mathbf{l}}^{\left( 2 \right)} \left( {\theta_{{}}^{\left( 2 \right)} } \right)^{3} \lambda^{4} ,{\text{A}}_{7902} = - \left( {\overline{f}_{24} + \overline{f}_{31} } \right)\left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\theta_{{}}^{\left( 2 \right)} \lambda \eta ,\;{\text{A}}_{8320} = \left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\overline{e}_{15} \theta_{{}}^{\left( 1 \right)} \lambda \; , $$
$$ {\text{A}}_{8302} = \left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\overline{e}_{24} \theta_{{}}^{\left( 1 \right)} \lambda \eta^{2} , $$
$$ {\text{A}}_{8410} = \left( {\overline{e}_{15} + \overline{e}_{31} } \right)\left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\theta_{{}}^{\left( 1 \right)} \lambda , $$
$$ {\text{A}}_{8501} = \left( {\overline{e}_{24} + \overline{e}_{31} } \right)\left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\theta_{{}}^{\left( 1 \right)} \lambda \eta , $$
$$ {\text{A}}_{8800} = - \left( {{{\pi^{2} } \mathord{\left/ {\vphantom {{\pi^{2} } {2\theta_{{}}^{\left( 1 \right)} }}} \right. \kern-\nulldelimiterspace} {2\theta_{{}}^{\left( 1 \right)} }}} \right)\overline{h}_{33}^{{}} ,\;{\text{A}}_{8820} = 0.5\theta_{{}}^{\left( 1 \right)} \overline{h}_{11} \lambda^{2} ,\;{\text{A}}_{8802} = 0.5\theta_{{}}^{\left( 1 \right)} \overline{h}_{22} \lambda^{2} \eta^{2} {\text{A}}_{9320} = \left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\overline{f}_{15} \theta_{{}}^{\left( 2 \right)} \lambda , $$
$$ {\text{A}}_{9302} = \left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\overline{f}_{24} \theta_{{}}^{\left( 2 \right)} \lambda \eta^{2} ,{\text{A}}_{9610} = \left( {\overline{f}_{15} + \overline{f}_{31} } \right)\left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\theta_{{}}^{\left( 2 \right)} \lambda ,{\text{A}}_{9701} = \left( {\overline{f}_{24} + \overline{f}_{31} } \right)\left( {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }} \right)\theta_{{}}^{\left( 2 \right)} \lambda \eta , $$
$$ {\text{A}}_{9900} = - \left( {{{\pi^{2} } \mathord{\left/ {\vphantom {{\pi^{2} } {2\theta_{{}}^{\left( 2 \right)} }}} \right. \kern-\nulldelimiterspace} {2\theta_{{}}^{\left( 2 \right)} }}} \right)\overline{\mu }_{33}^{{}} ,\;{\text{A}}_{9920} = 0.5\theta_{{}}^{\left( 2 \right)} \overline{\mu }_{11} \lambda^{2} , $$
in which,
$$ {\mathbf{a}}_{1104} = \left[ {0,\;\frac{8}{15},\;\frac{1}{4}} \right],{\mathbf{a}}_{1122} = \left[ {2,\;\frac{4}{3},\;\frac{1}{4}} \right],{\mathbf{a}}_{1213} = \left[ {2,\;\frac{4}{15},\; - \frac{1}{4}} \right],{\mathbf{a}}_{1440} = \left[ {1,\;\frac{2}{5},\;0} \right] , $$
$$ \begin{aligned}{\mathbf{a}}_{1404} &= \left[ {0,\;\frac{4}{15},\;\frac{1}{8}} \right],\,{\mathbf{a}}_{1422} = \left[ {1,\;\frac{2}{3},\;\frac{1}{8}} \right],\,{\mathbf{a}}_{1513} = \left[ {1,\;\frac{2}{15},\; - \frac{1}{8}} \right],\\ {\mathbf{a}}_{1640} &= \left[ { - 1,\; - \frac{2}{5},\;0} \right],\,{\mathbf{a}}_{1604} = \left[ {0,\; - \frac{4}{15},\; - \frac{1}{8}} \right], \end{aligned} $$
$$ {\mathbf{a}}_{3322} = - \left[ {0,\;\frac{16}{{15}},\;\frac{1}{2}} \right],\,{\mathbf{a}}_{3430} = \left[ {0,\; - \frac{16}{{15}},\;\frac{1}{4}} \right],\,{\mathbf{a}}_{4420} = - \left[ {2,\;\frac{32}{{15}},\;\frac{1}{4}} \right],\,{\mathbf{a}}_{4402} = - \left[ {0,\;\frac{4}{3},\;1} \right], $$
$$ {\mathbf{a}}_{4422} = \left[ {\frac{2}{3},\;\frac{4}{9},\;\frac{1}{12}} \right],\,{\mathbf{a}}_{4440} = \left[ {\frac{2}{3},\;\frac{4}{15},\;0} \right],\,{\mathbf{a}}_{4404} = \left[ {0,\;\frac{8}{45},\;\frac{1}{12}} \right],\,{\mathbf{a}}_{4511} = - \left[ {2,\;\frac{4}{5},\; - \frac{3}{4}} \right], $$
$$ {\mathbf{a}}_{4531} = \left[ {\frac{2}{3},\;\frac{4}{45},\; - \frac{1}{12}} \right]\,{\text{and}}\,{\mathbf{l}}^{\left( n \right)} = \left[ {\overline{c}_{44}^{\left( n \right)} \overline{l}_{0}^{2} ,\;\overline{c}_{44}^{\left( n \right)} \overline{l}_{1}^{2} ,\;\overline{c}_{44}^{\left( n \right)} \overline{l}_{2}^{2} } \right]^{T} . $$
Appendix D
For the PE-PM bi-layered composite plates with four simply supported edges, the displacements, deflection, rotations, electric and magnetic potentials can be expressed as
$$ \begin{gathered} \overline{u} = U\cos (\pi \overline{x}_{1}^{{}} )sin\left( {\pi \overline{x}_{2}^{{}} } \right),\;\overline{v} = V\sin (\pi \overline{x}_{1}^{{}} )\cos \left( {\pi \overline{x}_{2}^{{}} } \right),\;\overline{w} = W\sin (\pi \overline{x}_{1}^{{}} )sin\left( {\pi \overline{x}_{2}^{{}} } \right) , \hfill \\ \psi_{1}^{\left( 1 \right)} = \Psi_{1}^{\left( 1 \right)} \cos (\pi \overline{x}_{1}^{{}} )sin\left( {\pi \overline{x}_{2}^{{}} } \right),\;\psi_{2}^{\left( 1 \right)} = \Psi_{2}^{\left( 1 \right)} \sin (\pi \overline{x}_{1}^{{}} )\cos \left( {\pi \overline{x}_{2}^{{}} } \right),\;\psi_{1}^{\left( 2 \right)} = \Psi_{1}^{\left( 2 \right)} \cos (\pi \overline{x}_{1}^{{}} )sin\left( {\pi \overline{x}_{2}^{{}} } \right) , \hfill \\ \psi_{2}^{\left( 2 \right)} = \Psi_{2}^{\left( 2 \right)} \sin (\pi \overline{x}_{1}^{{}} )\cos \left( {\pi \overline{x}_{2}^{{}} } \right),\;\overline{\phi }^{\left( 1 \right)} = \Upsilon^{\left( 1 \right)} \sin \left( {\pi \overline{x}_{1}^{{}} } \right)sin\left( {\pi \overline{x}_{2}^{{}} } \right),\;\overline{\phi }^{\left( 2 \right)} = \Upsilon_{{}}^{\left( 2 \right)} \sin (\pi \overline{x}_{1}^{{}} )sin\left( {\pi \overline{x}_{2}^{{}} } \right) . \hfill \\ \end{gathered} $$
(43)
For the static bending problem, the applied transverse load \(\overline{q}\) can be also expanded in the double trigonometric series as
$$ \overline{q} = Q\sin \left( {\pi \overline{x}_{1} } \right)\sin \left( {\pi \overline{x}_{2} } \right) $$
(44)
where
$$ Q = 4\int_{0}^{1} {\int_{0}^{1} {\overline{q}\left( {\overline{x}_{1} ,\overline{x}_{2} } \right)} } \sin \left( {\pi \overline{x}_{1} } \right)\sin \left( {\pi \overline{x}_{2} } \right)d\overline{x}_{1} d\overline{x}_{2} . $$
(45)
In this paper thre special loading cases are discussed, i.e., sinusoidal distributed load (\(Q = q_{0}\)), uniformly distributed load (\(Q = {{16q_{0} } \mathord{\left/ {\vphantom {{16q_{0} } {\pi^{2} }}} \right. \kern-\nulldelimiterspace} {\pi^{2} }}\)), and point load (\(Q = 4q_{0}\)).
The following system of algebraic equations can be obtained as
$$ \left[ {\begin{array}{*{20}l} {K_{11} } \hfill & {K_{12} } \hfill & {K_{13} } \hfill & {K_{14} } \hfill & {K_{15} } \hfill & {K_{16} } \hfill & {K_{17} } \hfill & {K_{18} } \hfill & {K_{19} } \hfill \\ {K_{21} } \hfill & {K_{22} } \hfill & {K_{23} } \hfill & {K_{24} } \hfill & {K_{25} } \hfill & {K_{26} } \hfill & {K_{27} } \hfill & {K_{28} } \hfill & {K_{29} } \hfill \\ {K_{31} } \hfill & {K_{32} } \hfill & {K_{33} } \hfill & {K_{34} } \hfill & {K_{35} } \hfill & {K_{36} } \hfill & {K_{37} } \hfill & {K_{38} } \hfill & {K_{39} } \hfill \\ {K_{41} } \hfill & {K_{42} } \hfill & {K_{43} } \hfill & {K_{44} } \hfill & {K_{45} } \hfill & {K_{46} } \hfill & {K_{47} } \hfill & {K_{48} } \hfill & {K_{49} } \hfill \\ {K_{51} } \hfill & {K_{52} } \hfill & {K_{53} } \hfill & {K_{54} } \hfill & {K_{55} } \hfill & {K_{56} } \hfill & {K_{57} } \hfill & {K_{58} } \hfill & {K_{59} } \hfill \\ {K_{61} } \hfill & {K_{62} } \hfill & {K_{63} } \hfill & {K_{64} } \hfill & {K_{65} } \hfill & {K_{66} } \hfill & {K_{67} } \hfill & {K_{68} } \hfill & {K_{69} } \hfill \\ {K_{71} } \hfill & {K_{72} } \hfill & {K_{73} } \hfill & {K_{74} } \hfill & {K_{75} } \hfill & {K_{76} } \hfill & {K_{77} } \hfill & {K_{78} } \hfill & {K_{79} } \hfill \\ {K_{81} } \hfill & {K_{82} } \hfill & {K_{83} } \hfill & {K_{84} } \hfill & {K_{85} } \hfill & {K_{86} } \hfill & {K_{87} } \hfill & {K_{88} } \hfill & {K_{89} } \hfill \\ {K_{91} } \hfill & {K_{92} } \hfill & {K_{93} } \hfill & {K_{94} } \hfill & {K_{95} } \hfill & {K_{96} } \hfill & {K_{97} } \hfill & {K_{98} } \hfill & {K_{99} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} U \hfill \\ V \hfill \\ W \hfill \\ {\Psi_{1}^{\left( 1 \right)} } \hfill \\ {\Psi_{2}^{\left( 1 \right)} } \hfill \\ {\Psi_{1}^{\left( 2 \right)} } \hfill \\ {\Psi_{2}^{\left( 2 \right)} } \hfill \\ {\Upsilon^{\left( 1 \right)} } \hfill \\ {\Upsilon^{\left( 2 \right)} } \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} 0 \hfill \\ 0 \hfill \\ Q \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{array} } \right] $$
(46)
where
$$ \left\{ \begin{gathered} K_{11} = \left( { - 1} \right)\pi^{2} {\text{A}}_{1120} + \left( { - 1} \right)\pi^{2} {\text{A}}_{1102} + \pi^{4} {\text{A}}_{1140} + \pi^{4} {\text{A}}_{1104} + \pi^{4} {\text{A}}_{1122}, \; \hfill \\ K_{12} = \left( { - 1} \right)\pi^{2} {\text{A}}_{1211} + \pi^{4} {\text{A}}_{1213} + \pi^{4} {\text{A}}_{1231}, \; \hfill \\ K_{13} = 0,\;K_{14} = \left( { - 1} \right)\pi^{2} {\text{A}}_{1420} + \left( { - 1} \right)\pi^{2} {\text{A}}_{1402} + \pi^{4} {\text{A}}_{1440} + \pi^{4} {\text{A}}_{1404} + \pi^{4} {\text{A}}_{1422}, \; \hfill \\ K_{15} = \left( { - 1} \right)\pi^{2} {\text{A}}_{1511} + \pi^{4} {\text{A}}_{1513} + \pi^{4} {\text{A}}_{1531}, \hfill \\ K_{16} = \left( { - 1} \right)\pi^{2} {\text{A}}_{1620} + \left( { - 1} \right)\pi^{2} {\text{A}}_{1602} + \pi^{4} {\text{A}}_{1640} + \pi^{4} {\text{A}}_{1604} + \pi^{4} {\text{A}}_{1622}, \hfill \\ K_{17} = \left( { - 1} \right)\pi^{2} {\text{A}}_{1711} + \pi^{4} {\text{A}}_{1713} + \pi^{4} {\text{A}}_{1731} ,\;K_{18} = 0,\;K_{19} = 0, \hfill \\ \end{gathered} \right. $$
$$ \left\{ \begin{gathered} K_{21} = \left( { - 1} \right)\pi^{2} {\text{A}}_{2111} + \pi^{4} {\text{A}}_{2131} + \pi^{4} {\text{A}}_{2113}, \; \hfill \\ K_{22} = \left( { - 1} \right)\pi^{2} {\text{A}}_{2202} + \left( { - 1} \right)\pi^{2} {\text{A}}_{2220} + \pi^{4} {\text{A}}_{2204} + \pi^{4} {\text{A}}_{2240} + \pi^{4} {\text{A}}_{2222}, \hfill \\ K_{23} = 0,K_{24} = \left( { - 1} \right)\pi^{2} {\text{A}}_{2411} + \pi^{4} {\text{A}}_{2431} + \pi^{4} {\text{A}}_{2413}, \hfill \\ K_{25} = \left( { - 1} \right)\pi^{2} {\text{A}}_{2502} + \left( { - 1} \right)\pi^{2} {\text{A}}_{2520} + \pi^{4} {\text{A}}_{2504} + \pi^{4} {\text{A}}_{2540} + \pi^{4} {\text{A}}_{2522}, \hfill \\ K_{26} = \left( { - 1} \right)\pi^{2} {\text{A}}_{2611} + \pi^{4} {\text{A}}_{2631} + \pi^{4} {\text{A}}_{2613}, \hfill \\ K_{27} = \left( { - 1} \right)\pi^{2} {\text{A}}_{2702} + \left( { - 1} \right)\pi^{2} {\text{A}}_{2720} + \pi^{4} {\text{A}}_{2704} + \pi^{4} {\text{A}}_{2740} + \pi^{4} {\text{A}}_{2722} ,\;K_{28} = 0,\;K_{29} = 0 , \hfill \\ \end{gathered} \right. $$
$$ \left\{ \begin{gathered} K_{31} = 0,\;K_{32} = 0,\;K_{33} = \overline{k}_{w} + \left( { - 1} \right)\pi^{2} {\text{A}}_{3320} + \left( { - 1} \right)\pi^{2} {\text{A}}_{3302} + \pi^{4} {\text{A}}_{3340} + \pi^{4} {\text{A}}_{3304} + \pi^{4} {\text{A}}_{3322}, \hfill \\ K_{34} = \left( { - 1} \right)\pi {\text{A}}_{3410} + \pi^{3} {\text{A}}_{3430} + \pi^{3} {\text{A}}_{3412} ,\;K_{35} = \left( { - 1} \right)\pi {\text{A}}_{3501} + \pi^{3} {\text{A}}_{3521} + \pi^{3} {\text{A}}_{3503}, \hfill \\ K_{36} = \left( { - 1} \right)\pi {\text{A}}_{3610} + \pi^{3} {\text{A}}_{3630} + \pi^{3} {\text{A}}_{3612} ,\;K_{37} = \left( { - 1} \right)\pi {\text{A}}_{3701} + \pi^{3} {\text{A}}_{3721} + \pi^{3} {\text{A}}_{3703}, \hfill \\ K_{38} = \left( { - 1} \right)\pi^{2} {\text{A}}_{3820} + \left( { - 1} \right)\pi^{2} {\text{A}}_{3802} ,\;K_{39} = \left( { - 1} \right)\pi^{2} {\text{A}}_{3920} + \left( { - 1} \right)\pi^{2} {\text{A}}_{3902} , \hfill \\ \end{gathered} \right. $$
$$ \left\{ \begin{gathered} K_{41} = \left( { - 1} \right)\pi^{2} {\text{A}}_{4120} + \left( { - 1} \right)\pi^{2} {\text{A}}_{4102} + \pi^{4} {\text{A}}_{4140} + \pi^{4} {\text{A}}_{4104} + \pi^{4} {\text{A}}_{4122}, \hfill \\ K_{42} = \left( { - 1} \right)\pi^{2} {\text{A}}_{4211} + \pi^{4} {\text{A}}_{4213} + \pi^{4} {\text{A}}_{4231} ,\;K_{43} = \pi {\text{A}}_{4510} + \left( { - 1} \right)\pi^{3} {\text{A}}_{4530} + \left( { - 1} \right)\pi^{3} {\text{A}}_{4512}, \hfill \\ K_{44} = {\text{A}}_{4400} + \left( { - 1} \right)\pi^{2} {\text{A}}_{4420} + \left( { - 1} \right)\pi^{2} {\text{A}}_{4402} + \pi^{4} {\text{A}}_{4422} + \pi^{4} {\text{A}}_{4440} + \pi^{4} {\text{A}}_{4404}, \; \hfill \\ K_{45} = \left( { - 1} \right)\pi^{2} {\text{A}}_{4511} + \pi^{4} {\text{A}}_{4531} + \pi^{4} {\text{A}}_{4513} ,\;K_{46} = 0,\;K_{47} = 0,\;K_{48} = \pi {\text{A}}_{4810} ,\;K_{49} = 0 , \hfill \\ \end{gathered} \right. $$
$$ \left\{ \begin{gathered} K_{51} = \left( { - 1} \right)\pi^{2} {\text{A}}_{5111} + \left( { - 1} \right)\pi^{2} {\text{A}}_{5113} + \pi^{4} {\text{A}}_{5131}, \; \hfill \\ K_{52} = \left( { - 1} \right)\pi^{2} {\text{A}}_{5202} + \left( { - 1} \right)\pi^{2} {\text{A}}_{5220} + \pi^{4} {\text{A}}_{5222} + \pi^{4} {\text{A}}_{5204} + \pi^{4} {\text{A}}_{5240}, \hfill \\ K_{53} = \pi {\text{A}}_{5301} + \left( { - 1} \right)\pi^{3} {\text{A}}_{5303} + \left( { - 1} \right)\pi^{3} {\text{A}}_{5321} ,K_{54} = \left( { - 1} \right)\pi^{2} {\text{A}}_{5411} + \pi^{4} {\text{A}}_{5431} + \pi^{4} {\text{A}}_{5413}, \hfill \\ K_{55} = {\text{A}}_{5500} + \left( { - 1} \right)\pi^{2} {\text{A}}_{5502} + \left( { - 1} \right)\pi^{2} {\text{A}}_{5520} + \pi^{4} {\text{A}}_{5522} + \pi^{4} {\text{A}}_{5504} + \pi^{4} {\text{A}}_{5540}, \hfill \\ K_{56} = 0,\;K_{57} = 0,\;K_{58} = \pi {\text{A}}_{5801} ,K_{59} = 0 , \hfill \\ \end{gathered} \right. $$
$$ \left\{ \begin{gathered} K_{61} = \left( { - 1} \right)\pi^{2} {\text{A}}_{6120} + \left( { - 1} \right)\pi^{2} {\text{A}}_{6102} + \pi^{4} {\text{A}}_{6140} + \pi^{4} {\text{A}}_{6104} + \pi^{4} {\text{A}}_{6122}, \; \hfill \\ K_{62} = \left( { - 1} \right)\pi^{2} {\text{A}}_{6211} + \pi^{4} {\text{A}}_{6213} + \pi^{4} {\text{A}}_{6231} ,K_{63} = \pi {\text{A}}_{6310} + \left( { - 1} \right)\pi^{3} {\text{A}}_{6330} + \left( { - 1} \right)\pi^{3} {\text{A}}_{6312}, \; \hfill \\ K_{64} = 0,\;K_{65} = 0,\;K_{66} = {\text{A}}_{6600} + \left( { - 1} \right)\pi^{2} {\text{A}}_{6620} + \left( { - 1} \right)\pi^{2} {\text{A}}_{6602} + \pi^{4} {\text{A}}_{6622} + \pi^{4} {\text{A}}_{6640} + \pi^{4} {\text{A}}_{6604}, \; \hfill \\ K_{67} = \left( { - 1} \right)\pi^{2} {\text{A}}_{6711} + \pi^{4} {\text{A}}_{6731} + \pi^{4} {\text{A}}_{6713} ,\;K_{68} = 0,\;K_{69} = \pi {\text{A}}_{6910} , \hfill \\ \end{gathered} \right. $$
$$ \left\{ \begin{gathered} K_{71} = \left( { - 1} \right)\pi^{2} {\text{A}}_{7111} + \left( { - 1} \right)\pi^{2} {\text{A}}_{7113} + \pi^{4} {\text{A}}_{7131}, \hfill \\ K_{72} = \left( { - 1} \right)\pi^{2} {\text{A}}_{7202} + \left( { - 1} \right)\pi^{2} {\text{A}}_{7220} + \pi^{4} {\text{A}}_{7222} + \pi^{4} {\text{A}}_{7204} + \pi^{4} {\text{A}}_{7240}, \hfill \\ K_{73} = \pi {\text{A}}_{7301} + \left( { - 1} \right)\pi^{3} {\text{A}}_{7303} + \left( { - 1} \right)\pi^{3} {\text{A}}_{7321} ,K_{74} = 0,\;K_{75} = 0, \hfill \\ K_{76} = \left( { - 1} \right)\pi^{2} {\text{A}}_{7611} + \pi^{4} {\text{A}}_{7631} + \pi^{4} {\text{A}}_{7613}, \; \hfill \\ K_{77} = {\text{A}}_{7700} + \left( { - 1} \right)\pi^{2} {\text{A}}_{7702} + \left( { - 1} \right)\pi^{2} {\text{A}}_{7720} + \pi^{4} {\text{A}}_{7722} + \pi^{4} {\text{A}}_{7704} + \pi^{4} {\text{A}}_{7740}, \hfill \\ K_{78} = 0,K_{79} = \pi {\text{A}}_{7901} , \hfill \\ \end{gathered} \right. $$
$$ \left\{ \begin{gathered} K_{81} = 0,\;K_{82} = 0,\;K_{83} = \left( { - 1} \right)\pi^{2} {\text{A}}_{8320} + \left( { - 1} \right)\pi^{2} {\text{A}}_{8302} ,K_{84} = \left( { - 1} \right)\pi {\text{A}}_{8410} ,K_{85} = \left( { - 1} \right)\pi {\text{A}}_{8501}, \; \hfill \\ K_{86} = 0,\;K_{87} = 0,\;K_{88} = {\text{A}}_{8800} + \left( { - 1} \right)\pi^{2} {\text{A}}_{8820} + \left( { - 1} \right)\pi^{2} {\text{A}}_{8802} ,\;K_{89} = 0, \hfill \\ \end{gathered} \right. $$
$$ \left\{ \begin{gathered} K_{91} = 0,\;K_{92} = 0,\;K_{93} = \left( { - 1} \right)\pi^{2} {\text{A}}_{9320} + \left( { - 1} \right)\pi^{2} {\text{A}}_{9302} ,\;K_{94} = 0,\;K_{95} = 0, \hfill \\ K_{96} = \left( { - 1} \right)\pi {\text{A}}_{9610} ,K_{97} = \left( { - 1} \right)\pi {\text{A}}_{9701} ,K_{98} = 0,\;K_{99} = {\text{A}}_{9900} + \left( { - 1} \right)\pi^{2} {\text{A}}_{9920} + \left( { - 1} \right)\pi^{2} {\text{A}}_{9902} . \hfill \\ \end{gathered} \right. $$
