Appendix
The elements of the coefficient matrices in Eq. (25) are as follows:
$$ {\mathbf{A}}_{11}^{{\mathbf{k}}} = {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right) \, {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} ) - {\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - {\overline{\mathbf{C}}}_{66}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} - \frac{1}{{{\mathbf{R}}_{1} }}\left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} , $$
$$ {\mathbf{A}}_{12}^{{\mathbf{k}}} = - \left( {{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{66}^{{\mathbf{k}}} } \right){\mathbf{\alpha \beta }},\quad {\mathbf{A}}_{13}^{{\mathbf{k}}} = \left( {{\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} } \right){\mathbf{\alpha s}}^{{\mathbf{k}}} + \left\{ {\frac{{{\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} + \left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} } \right\}{{\varvec{\upalpha}}}, $$
$$ {\mathbf{A}}_{14}^{{\mathbf{k}}} = \left( {{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} } \right){\mathbf{\alpha s}}^{{\mathbf{k}}} + \left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} {{\varvec{\upalpha}}},\quad {\mathbf{A}}_{24}^{{\mathbf{k}}} = \left( {{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} } \right){\mathbf{\beta s}}^{{\mathbf{k}}} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{2}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} {{\varvec{\upbeta}}}, $$
$$ {\mathbf{A}}_{22}^{{\mathbf{k}}} = {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} ) - {\overline{\mathbf{C}}}_{66}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - {\overline{\mathbf{C}}}_{22}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} - \frac{1}{{{\mathbf{R}}_{2} }}\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{2}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} , $$
$$ {\mathbf{A}}_{23}^{{\mathbf{k}}} = \left( {{\overline{\mathbf{C}}}_{23}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} } \right){\mathbf{\beta s}}^{{\mathbf{k}}} + \left\{ {\frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{22}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{2}{{{\mathbf{R}}_{2} }}} \right) \, \mathop {\overline{C}}\nolimits_{44}^{k} } \right\}{{\varvec{\upbeta}}}, $$
$$ {\mathbf{A}}_{31}^{{\mathbf{k}}} = - \left( {{\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} } \right){\mathbf{\alpha s}}^{{\mathbf{k}}} + \left\{ {\frac{{{\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} + \frac{{{\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} } \right\}{{\varvec{\upalpha}}}, $$
$$ {\mathbf{A}}_{32}^{{\mathbf{k}}} = - \left( {{\overline{\mathbf{C}}}_{23}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} } \right){\mathbf{\beta s}}^{{\mathbf{k}}} + \left\{ {\frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{22}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} + \frac{{{\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} - \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{23}^{{\mathbf{k}}} } \right\}{{\varvec{\upbeta}}}, $$
$$ \begin{aligned} {\mathbf{A}}_{33}^{{\mathbf{k}}} & = {\overline{\mathbf{C}}}_{33}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{33}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} ) - {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} \\ & \quad { + }\left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right)\left( {\frac{{{\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{23}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right) - \frac{{{\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1}^{2} }} - 2\frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} {\mathbf{R}}_{2} }} - \frac{{{\overline{\mathbf{C}}}_{22}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2}^{2} }}} \right\}, \\ \end{aligned} $$
$$ {\mathbf{A}}_{34}^{{\mathbf{k}}} = {\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \frac{{{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{\mathbf{s}}^{{\mathbf{k}}} - {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - {\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} , $$
$$ {\mathbf{A}}_{41}^{{\mathbf{k}}} = - \left( {{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} } \right){\mathbf{\alpha s}}^{{\mathbf{k}}} + \left\{ {\frac{{{\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} } \right\}{{\varvec{\upalpha}}}, $$
$$ {\mathbf{A}}_{42}^{{\mathbf{k}}} = - \left( {{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} } \right){\mathbf{\beta s}}^{{\mathbf{k}}} + \left\{ {\frac{{{\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} }}{{R_{2} }} - \left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{2} }}} \right){\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} } \right\}{{\varvec{\upbeta}}}, $$
$$ \begin{aligned} {\mathbf{A}}_{43}^{{\mathbf{k}}} & = {\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} }}{{R_{1} }} - \frac{{{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} }}{{R_{2} }}} \right\}{\mathbf{s}}^{{\mathbf{k}}} , \\ & \quad + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right) \, \left( {\frac{{{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right) - {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - {\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} , \\ \end{aligned} $$
$$ {\mathbf{A}}_{44}^{{\mathbf{k}}} = - {\overline{\mathbf{\varepsilon }}}_{33}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} - \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{\varepsilon }}}_{33}^{{\mathbf{k}}} {\mathbf{s}}^{{\mathbf{k}}} + {\overline{\mathbf{\varepsilon }}}_{11}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} + {\overline{\mathbf{\varepsilon }}}_{22}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} $$
The elements of the coefficient matrices in Eq. (40) are given by
$$ \begin{aligned} {\overline{\mathbf{A}}}_{11}^{{\mathbf{k}}} & = {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} ) - {\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - 2{\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} {\mathbf{\alpha \beta }} \\ & \quad - {\overline{\mathbf{C}}}_{66}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} - \frac{1}{{{\mathbf{R}}_{1} }}\left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} , \\ \end{aligned} $$
$$ {\overline{\mathbf{A}}}_{12}^{{\mathbf{k}}} = {\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} + 2\frac{{{\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }}{\mathbf{s}}^{{\mathbf{k}}} - {\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - \left( {{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{66}^{{\mathbf{k}}} } \right){\mathbf{\alpha \beta }} - {\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} - \frac{1}{{{\mathbf{R}}_{2} }}\left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} , $$
$$ \begin{aligned} {\overline{\mathbf{A}}}_{13}^{{\mathbf{k}}} & = \left\{ {\left( {{\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} + \left( {{\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}{\mathbf{s}}^{{\mathbf{k}}} + \left\{ {\frac{{{\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} + \left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} } \right\}{{\varvec{\upalpha}}} \\ & \quad + \left\{ {\frac{{{\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} + \left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} } \right\}{{\varvec{\upbeta}}}, \\ \end{aligned} $$
$$ {\overline{\mathbf{A}}}_{14}^{{\mathbf{k}}} = \left\{ {\left( {{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} + \left( {{\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{25}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}{\mathbf{s}}^{{\mathbf{k}}} + \left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right)\left( {{\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} {{\varvec{\upalpha}}} + {\overline{\mathbf{e}}}_{25}^{{\mathbf{k}}} {{\varvec{\upbeta}}}} \right), $$
$$ \begin{aligned} {\overline{\mathbf{A}}}_{21}^{{\mathbf{k}}} & = {\overline{\mathbf{A}}}_{12}^{{\mathbf{k}}} ,\;{\overline{\mathbf{A}}}_{22}^{{\mathbf{k}}} = {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} ) - {\overline{\mathbf{C}}}_{66}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - 2{\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} {\mathbf{\alpha \beta }} \\ & \quad - {\overline{\mathbf{C}}}_{22}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} - \frac{1}{{{\mathbf{R}}_{2} }}\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{2}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} , \\ \end{aligned} $$
$$ \begin{aligned} {\overline{\mathbf{A}}}_{23}^{{\mathbf{k}}} & = \left\{ {\left( {{\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} + \left( {{\overline{\mathbf{C}}}_{23}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}s^{k} \\ & \quad + \left\{ {\frac{{{\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{2}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} } \right\}{{\varvec{\upalpha}}} + \left\{ {\frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{22}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{2}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} } \right\}{{\varvec{\upbeta}}}, \\ \end{aligned} $$
$$ {\overline{\mathbf{A}}}_{24}^{{\mathbf{k}}} = \left\{ {\left( {{\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} + \left( {{\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}{\mathbf{s}}^{{\mathbf{k}}} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{2}{{{\mathbf{R}}_{2} }}} \right)\left( {{\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} {{\varvec{\upalpha}}} + {\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} {{\varvec{\upbeta}}}} \right), $$
$$ \begin{aligned} {\overline{\mathbf{A}}}_{31}^{{\mathbf{k}}} & = \left\{ {\left( {{\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} + \left( {{\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}{\mathbf{s}}^{{\mathbf{k}}} \\ & \quad + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} - \frac{{{\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }}} \right\}{{\varvec{\upalpha}}} \\ & \quad + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \frac{{{\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} - \frac{{{\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }}} \right\}{{\varvec{\upbeta}}}, \\ \end{aligned} $$
$$ \begin{aligned} {\overline{\mathbf{A}}}_{32}^{{\mathbf{k}}} & = \left\{ {\left( {{\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} + \left( {{\overline{\mathbf{C}}}_{23}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}{\mathbf{s}}^{{\mathbf{k}}} \\ & \quad + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \frac{{{\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} - \frac{{{\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{{\varvec{\upalpha}}} \\ & \quad + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{23}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \frac{{{\overline{\mathbf{C}}}_{22}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} - \frac{{{\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{{\varvec{\upbeta}}}, \\ \end{aligned} $$
$$ \begin{aligned} {\overline{\mathbf{A}}}_{33}^{{\mathbf{k}}} & = {\overline{\mathbf{C}}}_{33}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{33}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} ) + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right)\left( {\frac{{{\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{23}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right) \\ & \quad - \frac{{{\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1}^{2} }} - 2\frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} {\mathbf{R}}_{2} }} - \frac{{{\overline{\mathbf{C}}}_{22}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2}^{2} }} - {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - 2{\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} {\mathbf{\alpha \beta }} - {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} , \\ \end{aligned} $$
$$ \begin{aligned} {\overline{\mathbf{A}}}_{34}^{{\mathbf{k}}} & = {\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} - {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - \left( {{\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} } \right){\mathbf{\alpha \beta }} - {\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} \\ & \quad + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \frac{{{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{\mathbf{s}}^{{\mathbf{k}}} , \\ \end{aligned} $$
$$ {\overline{\mathbf{A}}}_{41}^{{\mathbf{k}}} = \left\{ {({\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} {)}{{\varvec{\upalpha}}} + ({\overline{\mathbf{e}}}_{25}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} {)}{{\varvec{\upbeta}}}} \right\}{\mathbf{s}}^{{\mathbf{k}}} + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }}} \right\}{{\varvec{\upalpha}}} + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} - \frac{{\mathop {\overline{e}}\nolimits_{25}^{k} }}{{{\mathbf{R}}_{1} }}} \right\}{{\varvec{\upbeta}}}, $$
$$ {\overline{\mathbf{A}}}_{42}^{{\mathbf{k}}} = \left\{ {\left( {{\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} + \left( {{\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}{\mathbf{s}}^{{\mathbf{k}}} + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{{\varvec{\upalpha}}} + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{{\varvec{\upbeta}}}, $$
$$ \begin{aligned} {\overline{\mathbf{A}}}_{43}^{{\mathbf{k}}} & = {\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} + \frac{{{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{\mathbf{s}}^{{\mathbf{k}}} \\ & \quad - \, {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - \left( {{\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{25}^{{\mathbf{k}}} } \right){\mathbf{\alpha \beta }} - {\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} \, \\ & \quad + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right) \, \left( {\frac{{{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right), \\ \end{aligned} $$
$$ {\overline{\mathbf{A}}}_{44}^{{\mathbf{k}}} = - {\overline{\mathbf{\varepsilon }}}_{33}^{{\mathbf{k}}} ({\mathbf{s}}^{{\mathbf{k}}} )^{2} - \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{\varepsilon }}}_{33}^{{\mathbf{k}}} {\mathbf{s}}^{{\mathbf{k}}} + {\overline{\mathbf{\varepsilon }}}_{11}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} + 2{\overline{\mathbf{\varepsilon }}}_{12}^{{\mathbf{k}}} {\mathbf{\alpha \beta }} + {\overline{\mathbf{\varepsilon }}}_{22}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} , $$
$$ {\mathbf{B}}_{11}^{{\mathbf{k}}} = {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} ({\mathbf{r}}^{{\mathbf{k}}} )^{2} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} ({\mathbf{r}}^{{\mathbf{k}}} ) - {\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} + 2{\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} {\mathbf{\alpha \beta }} - {\overline{\mathbf{C}}}_{66}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} - \frac{1}{{{\mathbf{R}}_{1} }}\left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} , $$
$$ {\mathbf{B}}_{12}^{{\mathbf{k}}} = {\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} ({\mathbf{r}}^{{\mathbf{k}}} )^{2} + 2\frac{{{\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }}{\mathbf{r}}^{{\mathbf{k}}} - {\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} + \left( {{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{66}^{{\mathbf{k}}} } \right){\mathbf{\alpha \beta }} - {\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} - \frac{1}{{{\mathbf{R}}_{2} }}\left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} , $$
$$ \begin{aligned} {\mathbf{B}}_{13}^{{\mathbf{k}}} & = \left\{ {\left( {{\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} - \left( {{\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}{\mathbf{r}}^{{\mathbf{k}}} + \left( {\frac{{{\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right){{\varvec{\upalpha}}}, \\ & \quad - \left( {\frac{{{\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right){{\varvec{\upbeta}}} + \left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right)\left( {{\overline{\mathbf{C}}}_{15}^{{\mathbf{k}}} {{\varvec{\upalpha}}} - {\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} {{\varvec{\upbeta}}}} \right){,} \\ \end{aligned} $$
$$ {\mathbf{B}}_{14}^{{\mathbf{k}}} = \left\{ {\left( {{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} { + }{\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} - \left( {{\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} { + }{\overline{\mathbf{e}}}_{25}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}{\mathbf{r}}^{{\mathbf{k}}} + \left( {\frac{2}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right)\left( {{\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} {{\varvec{\upalpha}}} + {\overline{\mathbf{e}}}_{25}^{{\mathbf{k}}} {{\varvec{\upbeta}}}} \right), $$
$$ \begin{aligned} {\mathbf{B}}_{21}^{{\mathbf{k}}} & = {\mathbf{B}}_{12}^{{\mathbf{k}}} ,{\mathbf{B}}_{22}^{{\mathbf{k}}} = {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} ({\mathbf{r}}^{{\mathbf{k}}} )^{2} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} ({\mathbf{r}}^{{\mathbf{k}}} ) - {\overline{\mathbf{C}}}_{66}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} + 2{\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} {\mathbf{\alpha \beta }} \\ & \quad - {\overline{\mathbf{C}}}_{22}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} - \frac{1}{{{\mathbf{R}}_{2} }}\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{2}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} , \\ \end{aligned} $$
$$ \begin{aligned} {\mathbf{B}}_{23}^{{\mathbf{k}}} & = \left( {\frac{{{\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right){{\varvec{\upalpha}}} - \left( {\frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{22}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right){{\varvec{\upbeta}}} + \left( {{\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} } \right){\mathbf{\alpha r}}^{{\mathbf{k}}} - \left( {{\overline{\mathbf{C}}}_{23}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} } \right){\mathbf{\beta r}}^{{\mathbf{k}}} , \\ & \quad + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{2}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} {{\varvec{\upalpha}}} - \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{2}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} {{\varvec{\upbeta}}}, \\ \end{aligned} $$
$$ {\mathbf{B}}_{24}^{{\mathbf{k}}} = \left\{ {\left( {{\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} - \left( {{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}{\mathbf{r}}^{{\mathbf{k}}} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{2}{{{\mathbf{R}}_{2} }}} \right)\left( {{\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} {{\varvec{\upalpha}}} - {\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} {{\varvec{\upbeta}}}} \right){, } $$
$$ \begin{aligned} {\mathbf{B}}_{31}^{{\mathbf{k}}} & = \left\{ {\left( {{\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} - \left( {{\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}r^{k} \\ & \quad + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} - \frac{{{\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }}} \right\}{{\varvec{\upalpha}}} \\ & \quad + \left\{ {\frac{{{\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} + \frac{{{\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} } \right\}{{\varvec{\upbeta}}}, \\ \end{aligned} $$
$$ \begin{aligned} \mathop { \, B}\nolimits_{32}^{k} & = \left\{ {\left( {{\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} } \right){{\varvec{\upalpha}}} - \left( {{\overline{\mathbf{C}}}_{23}^{{\mathbf{k}}} + {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} } \right){{\varvec{\upbeta}}}} \right\}r^{k} \\ & \quad + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \frac{{{\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} - \frac{{{\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{{\varvec{\upalpha}}} \\ & \quad + \left\{ {\frac{{{\overline{\mathbf{C}}}_{16}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{26}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} + \frac{{{\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }} - \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{36}^{{\mathbf{k}}} } \right\}{{\varvec{\upbeta}}}, \\ \end{aligned} $$
$$ \begin{aligned} {\mathbf{B}}_{33}^{{\mathbf{k}}} & = {\overline{\mathbf{C}}}_{33}^{{\mathbf{k}}} ({\mathbf{r}}^{{\mathbf{k}}} )^{2} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{C}}}_{33}^{{\mathbf{k}}} ({\mathbf{r}}^{{\mathbf{k}}} ) + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right)\left( {\frac{{{\overline{\mathbf{C}}}_{13}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{C}}}_{23}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right) \\ & \quad - \frac{{{\overline{\mathbf{C}}}_{11}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1}^{2} }} - 2\frac{{{\overline{\mathbf{C}}}_{12}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} {\mathbf{R}}_{2} }} - \frac{{{\overline{\mathbf{C}}}_{22}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2}^{2} }} - {\overline{\mathbf{C}}}_{55}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} + 2{\overline{\mathbf{C}}}_{45}^{{\mathbf{k}}} {\mathbf{\alpha \beta }} - {\overline{\mathbf{C}}}_{44}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} , \\ \end{aligned} $$
$$ {\mathbf{B}}_{34}^{{\mathbf{k}}} = {\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} ({\mathbf{r}}^{{\mathbf{k}}} )^{2} - {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} + \left( {{\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{25}^{{\mathbf{k}}} } \right){\mathbf{\alpha \beta }} - {\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} - \frac{{{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{\mathbf{r}}^{{\mathbf{k}}} , $$
$$ \begin{aligned} {\mathbf{B}}_{41}^{{\mathbf{k}}} & = \left\{ {({\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} {)}{{\varvec{\upalpha}}} - ({\overline{\mathbf{e}}}_{25}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} {)}{{\varvec{\upbeta}}}} \right\}r^{k} + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }}} \right\}{{\varvec{\upalpha}}} \\ & \quad - \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{e}}}_{25}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{{\varvec{\upbeta}}}, \\ \end{aligned} $$
$$ \begin{aligned} {\mathbf{B}}_{42}^{{\mathbf{k}}} & = \left\{ {({\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} {)}{{\varvec{\upalpha}}} - ({\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} {)}{{\varvec{\upbeta}}}} \right\}{\mathbf{r}}^{{\mathbf{k}}} + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{36}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{{\varvec{\upalpha}}} \\ & \quad - \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} - \frac{{{\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{{\varvec{\upbeta}}}, \\ \end{aligned} $$
$$ \begin{aligned} {\mathbf{B}}_{43}^{{\mathbf{k}}} & = {\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} ({\mathbf{r}}^{{\mathbf{k}}} )^{2} + \left\{ {\left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{e}}}_{33}^{{\mathbf{k}}} + \frac{{{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right\}{\mathbf{r}}^{{\mathbf{k}}} + \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right) \, \left( {\frac{{{\overline{\mathbf{e}}}_{31}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{1} }} + \frac{{{\overline{\mathbf{e}}}_{32}^{{\mathbf{k}}} }}{{{\mathbf{R}}_{2} }}} \right) \\ & \quad - {\overline{\mathbf{e}}}_{15}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - {\overline{\mathbf{e}}}_{24}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} + \left( {{\overline{\mathbf{e}}}_{14}^{{\mathbf{k}}} + {\overline{\mathbf{e}}}_{25}^{{\mathbf{k}}} } \right){\mathbf{\alpha \beta }}, \\ \end{aligned} $$
$$ {\mathbf{B}}_{44}^{{\mathbf{k}}} = - {\overline{\mathbf{\varepsilon }}}_{33}^{{\mathbf{k}}} ({\mathbf{r}}^{{\mathbf{k}}} )^{2} - \left( {\frac{1}{{{\mathbf{R}}_{1} }} + \frac{1}{{{\mathbf{R}}_{2} }}} \right){\overline{\mathbf{\varepsilon }}}_{33}^{{\mathbf{k}}} {\mathbf{r}}^{{\mathbf{k}}} + {\overline{\mathbf{\varepsilon }}}_{11}^{{\mathbf{k}}} {{\varvec{\upalpha}}}^{2} - 2{\overline{\mathbf{\varepsilon }}}_{12}^{{\mathbf{k}}} {\mathbf{\alpha \beta }} +{\overline{\mathbf{\varepsilon }}}_{22}^{{\mathbf{k}}} {{\varvec{\upbeta}}}^{2} $$
