Appendix 1
The expression of each element in Eq. (1)
$$\left. \begin{gathered} a_{11} = \frac{{A_{13} A_{22} - A_{12} A_{23} }}{{A_{11} A_{22} - A_{12} A_{21} }},a_{12} = \frac{{A_{14} A_{22} }}{{A_{11} A_{22} - A_{12} A_{21} }},a_{13} = - \frac{{A_{12} A_{24} }}{{A_{11} A_{22} - A_{12} A_{21} }} \hfill \\ a_{21} = \frac{{A_{13} A_{21} - A_{11} A_{23} }}{{A_{12} A_{21} - A_{11} A_{22} }},a_{22} = \frac{{A_{14} A_{21} }}{{A_{12} A_{21} - A_{11} A_{22} }},a_{23} = - \frac{{A_{11} A_{24} }}{{A_{12} A_{21} - A_{11} A_{22} }} \hfill \\ \end{gathered} \right\},\vartheta^{l} = \frac{{nS_{l} }}{{{\mathbf{K}}_{l} }},\vartheta^{g} = \frac{{nS_{g} }}{{{\mathbf{K}}_{g} }},$$
\(B_{1} = \alpha S_{{\text{e}}} a_{12} + \alpha \left( {1 - S_{{\text{e}}} } \right)a_{22}\), \(B_{2} = \alpha S_{{\text{e}}} a_{13} + \alpha \left( {1 - S_{{\text{e}}} } \right)a_{23}\), \(\lambda_{{\text{c}}} = \lambda + \alpha S_{{\text{e}}} a_{11} + \alpha \left( {1 - S_{{\text{e}}} } \right)a_{21}\).
with
$$\left. \begin{gathered} A_{11} = \xi S_{ww} S_{l} + \frac{{nS_{l} }}{{K^{w} }} - n\frac{{\partial S_{l} }}{{\partial p_{{\text{c}}} }},A_{12} = \xi S_{gg} S_{l} + n\frac{{\partial S_{l} }}{{\partial p_{{\text{c}}} }}A_{13} = \alpha S_{l} ,A_{14} = nS_{l} \hfill \\ A_{21} = \xi S_{ww} S_{g} + n\frac{{\partial S_{l} }}{{\partial p_{{\text{c}}} }},A_{22} = \xi S_{gg} S_{g} + \frac{{nS_{g} }}{{K^{g} }} - n\frac{{\partial S_{l} }}{{\partial p_{{\text{c}}} }},A_{23} = \alpha S_{g} ,A_{24} = nS_{g} \hfill \\ \end{gathered} \right\},$$
\(\alpha = 1 - {{K_{b} } \mathord{\left/ {\vphantom {{K_{b} } {K_{s} }}} \right. \kern-0pt} {K_{s} }}\), \(S_{e} = {{\left( {S_{l} - S_{res}^{l} } \right)} \mathord{\left/ {\vphantom {{\left( {S_{l} - S_{res}^{l} } \right)} {\left( {S_{sat}^{l} - S_{res}^{l} } \right)}}} \right. \kern-0pt} {\left( {S_{sat}^{l} - S_{res}^{l} } \right)}}\), \(K_{l} = {{k_{rl} } \mathord{\left/ {\vphantom {{k_{rl} } {\mu_{l} }}} \right. \kern-0pt} {\mu_{l} }}\), \(K_{g} = {{k_{rg} } \mathord{\left/ {\vphantom {{k_{rg} } {\mu_{g} }}} \right. \kern-0pt} {\mu_{g} }}\),
\(k_{r}^{l} = \sqrt {S_{e} } \left[ {1 - \left( {1 - S_{e}^{{{1 \mathord{\left/ {\vphantom {1 m}} \right. \kern-0pt} m}}} } \right)^{m} } \right]^{2}\), \(k_{r}^{g} = \sqrt {1 - S_{e} } \left( {1 - S_{e}^{{{1 \mathord{\left/ {\vphantom {1 m}} \right. \kern-0pt} m}}} } \right)^{2m}\).
where \(\chi\), m and \(d\) are the material parameters of the V-G model. \(S_{e}\) is the effective saturation of the pore fluid with \(S_{res}^{l}\) and \(S_{sat}^{l}\) the residual saturation and complete saturation, respectively. This paper considers \(S_{sat}^{l} = 1\). Other symbols are expressed as
\(\xi { = }{{\left( {\alpha - n} \right)} \mathord{\left/ {\vphantom {{\left( {\alpha - n} \right)} {K_{s} }}} \right. \kern-0pt} {K_{s} }}\), \(p_{c} = \left( {{1 \mathord{\left/ {\vphantom {1 \chi }} \right. \kern-0pt} \chi }} \right)\rho_{s} + \left( {S_{e}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} m}} \right. \kern-0pt} m}}} - 1} \right)^{{{1 \mathord{\left/ {\vphantom {1 d}} \right. \kern-0pt} d}}}\), \(S_{ww} = S_{e} + {{p_{c} } \mathord{\left/ {\vphantom {{p_{c} } {\left( {S_{sat} - S_{res} } \right)}}} \right. \kern-0pt} {\left( {S_{sat} - S_{res} } \right)}}A_{s}\), \(S_{gg} = \left( {1 - S_{e} } \right) - {{p_{c} } \mathord{\left/ {\vphantom {{p_{c} } {\left( {S_{sat} - S_{res} } \right)}}} \right. \kern-0pt} {\left( {S_{sat} - S_{res} } \right)}}A_{s}\), \(A_{s} = \chi md\left( {S_{sat} - S_{res} } \right)S_{e}^{{{{\left( {m + 1} \right)} \mathord{\left/ {\vphantom {{\left( {m + 1} \right)} m}} \right. \kern-0pt} m}}} \left( {S_{e}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} m}} \right. \kern-0pt} m}}} - 1} \right)^{{{{\left( {d - 1} \right)} \mathord{\left/ {\vphantom {{\left( {d - 1} \right)} d}} \right. \kern-0pt} d}}}\).
Appendix 2
The expression of each element in Eqs. (5a) and (5b)
$$\left. \begin{gathered} b_{11} = \rho^{l} \omega^{2} - a_{11} k_{{\text{P}}}^{2} ,b_{12} = \rho^{l} \omega^{2} + \vartheta^{l} {\text{i}} \omega - a_{12} k_{{\text{P}}}^{2} ,b_{13} = - a_{13} k_{{\text{P}}}^{2} ,b_{21} = \rho^{g} \omega^{2} - a_{21} k_{{\text{P}}}^{2} \hfill \\ b_{23} = \rho^{g} \omega^{2} + \vartheta^{g} {\text{i}} \omega - a_{23} k_{{\text{P}}}^{2} ,b_{31} = \rho \omega^{2} - \left( {\lambda_{c} + 2\mu } \right)k_{{\text{P}}}^{2} ,b_{32} = nS^{l} \rho^{l} \omega^{2} - B_{1} k_{{\text{P}}}^{2} ,b_{33} = nS^{g} \rho^{g} \omega^{2} - B_{2} k_{{\text{P}}}^{2} \hfill \\ \end{gathered} \right\},$$
$$\left. \begin{gathered} c_{11} = \rho^{l} \omega^{2} ,c_{12} = \rho^{l} \omega^{2} + \vartheta^{l} {\text{i}} \omega ,c_{13} = 0,c_{21} = \rho^{g} \omega^{2} ,c_{12} = \rho^{l} \omega^{2} + \vartheta^{l} {\text{i}} \omega ,c_{13} = 0 \hfill \\ c_{21} = \rho^{g} \omega^{2} ,c_{22} = 0,c_{23} = \rho^{g} \omega^{2} + \vartheta^{g} {\text{i}} \omega ,c_{31} = \rho \omega^{2} - \mu k_{{\text{S}}}^{2} ,c_{32} = nS^{l} \rho^{l} \omega^{2} ,c_{33} = nS^{g} \rho^{g} \omega^{2} \hfill \\ \end{gathered} \right\}.$$
Appendix 3
The expression of each element in Eq. (16)
$$\left. \begin{gathered} f_{11} = \left[ { - \left( {\lambda_{c}^{\rm I} + 2\mu^{\rm I} n_{{{\text{P}}_{1} }}^{{{\text{re2}}}} } \right) - B_{1}^{\rm I} \delta_{{l{\text{P}}_{1} }}^{{{\text{re}}}} - B_{2}^{\rm I} \delta_{{g{\text{P}}_{1} }}^{{{\text{re}}}} } \right]k_{{{\text{P}}_{1} }}^{{{\text{re2}}}} ,f_{12} = \left[ { - \left( {\lambda_{c}^{\rm I} + 2\mu^{\rm I} n_{{{\text{P}}_{2} }}^{{{\text{re2}}}} } \right) - B_{1}^{\rm I} \delta_{{l{\text{P}}_{2} }}^{{{\text{re}}}} - B_{2}^{\rm I} \delta_{{g{\text{P}}_{2} }}^{{{\text{re}}}} } \right]k_{{{\text{P}}_{2} }}^{{{\text{re2}}}} \hfill \\ f_{13} = \left[ { - \left( {\lambda_{c}^{\rm I} A_{{s{\text{P}}_{3} }}^{{{\text{re}}}} + 2\mu^{\rm I} n_{{{\text{P}}_{3} }}^{{{\text{re2}}}} } \right) - B_{1}^{\rm I} \delta_{{l{\text{P}}_{3} }}^{{{\text{re}}}} - B_{2}^{\rm I} \delta_{{g{\text{P}}_{3} }}^{{{\text{re}}}} } \right]k_{{{\text{P}}_{3} }}^{{{\text{re2}}}} ,f_{14} = \left( {\lambda_{c}^{{{\rm I}{\rm I}}} + 2\mu^{{{\rm I}{\rm I}}} n_{{{\text{P}}_{1} }}^{{{\text{tr2}}}} + B_{1}^{{{\rm I}{\rm I}}} \delta_{{l{\text{P}}_{1} }}^{{{\text{tr}}}} + B_{2}^{{{\rm I}{\rm I}}} \delta_{{g{\text{P}}_{1} }}^{{{\text{tr}}}} } \right)k_{{{\text{P}}_{1} }}^{{{\text{tr2}}}} \hfill \\ f_{15} = \left( {\lambda_{c}^{{{\rm I}{\rm I}}} + 2\mu^{{{\rm I}{\rm I}}} n_{{{\text{P}}_{2} }}^{{{\text{tr2}}}} + B_{1}^{{{\rm I}{\rm I}}} \delta_{{l{\text{P}}_{2} }}^{{{\text{tr}}}} + B_{2}^{{{\rm I}{\rm I}}} \delta_{{g{\text{P}}_{2} }}^{{{\text{tr}}}} } \right)k_{{{\text{P}}_{2} }}^{{{\text{tr2}}}} ,f_{16} = \left( {\lambda_{c}^{{{\rm I}{\rm I}}} + 2\mu^{{{\rm I}{\rm I}}} n_{{{\text{P}}_{3} }}^{{{\text{tr2}}}} + B_{1}^{{{\rm I}{\rm I}}} \delta_{{l{\text{P}}_{3} }}^{{{\text{tr}}}} + B_{2}^{{{\rm I}{\rm I}}} \delta_{{g{\text{P}}_{3} }}^{{{\text{tr}}}} } \right)k_{{{\text{P}}_{3} }}^{{{\text{tr2}}}} \hfill \\ f_{17} = - 2\mu^{\rm I} l_{{\text{S}}}^{{{\text{re}}}} n_{{\text{S}}}^{{{\text{re}}}} k_{{\text{S}}}^{{{\text{re2}}}} ,f_{18} = - 2\mu^{{{\rm I}{\rm I}}} l_{{\text{S}}}^{{{\text{tr}}}} n_{{\text{S}}}^{{{\text{tr}}}} k_{{\text{S}}}^{{{\text{tr2}}}} \hfill \\ \end{gathered} \right\},$$
$$\left. \begin{gathered} f_{21} = - 2\mu^{\rm I} l_{{{\text{P}}_{1} }}^{{{\text{re}}}} n_{{{\text{P}}_{1} }}^{{{\text{re}}}} k_{{{\text{P}}_{1} }}^{{{\text{re2}}}} ,f_{22} = - 2\mu^{\rm I} l_{{{\text{P}}_{2} }}^{{{\text{re}}}} n_{{{\text{P}}_{2} }}^{{{\text{re}}}} k_{{{\text{P}}_{2} }}^{{{\text{re2}}}} ,f_{23} = - 2\mu^{\rm I} l_{{{\text{P}}_{3} }}^{{{\text{re}}}} n_{{{\text{P}}_{3} }}^{{{\text{re}}}} k_{{{\text{P}}_{3} }}^{{{\text{re2}}}} ,f_{24} = - 2\mu^{{{\rm I}{\rm I}}} l_{{{\text{P}}_{1} }}^{{{\text{tr}}}} n_{{{\text{P}}_{1} }}^{{{\text{tr}}}} k_{{{\text{P}}_{1} }}^{{{\text{tr2}}}} \hfill \\ f_{25} = - 2\mu^{{{\rm I}{\rm I}}} l_{{{\text{P}}_{2} }}^{{{\text{tr}}}} n_{{{\text{P}}_{2} }}^{{{\text{tr}}}} k_{{{\text{P}}_{2} }}^{{{\text{tr2}}}} ,f_{26} = - 2\mu^{{{\rm I}{\rm I}}} l_{{{\text{P}}_{3} }}^{{{\text{tr}}}} n_{{{\text{P}}_{3} }}^{{{\text{tr}}}} k_{{{\text{P}}_{3} }}^{{{\text{tr2}}}} ,f_{27} = \mu^{\rm I} \left( {n_{{\text{S}}}^{{{\text{re2}}}} - l_{{\text{S}}}^{{{\text{re2}}}} } \right)k_{{\text{S}}}^{{{\text{re2}}}} ,f_{28} = \mu^{{{\rm I}{\rm I}}} \left( {l_{{\text{S}}}^{{{\text{tr2}}}} - n_{{\text{S}}}^{{{\text{tr2}}}} } \right)k_{{\text{S}}}^{{{\text{tr2}}}} \hfill \\ \end{gathered} \right\},$$
$$f_{31} = n_{{{\text{P}}_{1} }}^{{{\text{re}}}} k_{{{\text{P}}_{1} }}^{{{\text{re}}}} ,f_{32} = n_{{{\text{P}}_{2} }}^{{{\text{re}}}} k_{{{\text{P}}_{2} }}^{{{\text{re}}}} ,f_{33} = n_{{{\text{P}}_{3} }}^{{{\text{re}}}} k_{{{\text{P}}_{3} }}^{{{\text{re}}}} ,f_{34} = n_{{{\text{P}}_{1} }}^{{{\text{tr}}}} k_{{{\text{P}}_{1} }}^{{{\text{tr}}}} ,f_{35} = n_{{{\text{P}}_{2} }}^{{{\text{tr}}}} k_{{{\text{P}}_{2} }}^{{{\text{tr}}}} ,f_{36} = n_{{{\text{P}}_{3} }}^{{{\text{tr}}}} k_{{{\text{P}}_{3} }}^{{{\text{tr}}}} ,f_{37} = l_{{\text{S}}}^{{{\text{re}}}} k_{{\text{S}}}^{{{\text{re}}}} ,f_{38} = - l_{{\text{S}}}^{{{\text{tr}}}} k_{{\text{S}}}^{{{\text{tr}}}} ,$$
$$f_{41} = l_{{{\text{P}}_{1} }}^{{{\text{re}}}} k_{{{\text{P}}_{1} }}^{{{\text{re}}}} ,f_{42} = l_{{{\text{P}}_{2} }}^{{{\text{re}}}} k_{{{\text{P}}_{2} }}^{{{\text{re}}}} ,f_{43} = l_{{{\text{P}}_{3} }}^{{{\text{re}}}} k_{{{\text{P}}_{3} }}^{{{\text{re}}}} ,f_{44} = - l_{{{\text{P}}_{1} }}^{{{\text{tr}}}} k_{{{\text{P}}_{1} }}^{{{\text{tr}}}} ,f_{45} = - l_{{{\text{P}}_{2} }}^{{{\text{tr}}}} k_{{{\text{P}}_{2} }}^{{{\text{tr}}}} ,f_{46} = - l_{{{\text{P}}_{3} }}^{{{\text{tr}}}} k_{{{\text{P}}_{3} }}^{{{\text{tr}}}} ,f_{47} = - n_{{\text{S}}}^{{{\text{re}}}} k_{{\text{S}}}^{{{\text{re}}}} ,f_{48} = - n_{{\text{S}}}^{{{\text{tr}}}} k_{{\text{S}}}^{{{\text{tr}}}} ,$$
$$f_{51} = n_{{{\text{P}}_{1} }}^{{{\text{re}}}} \delta_{{l{\text{P}}_{1} }}^{{{\text{re}}}} k_{{{\text{P}}_{1} }}^{{{\text{re}}}} ,f_{52} = n_{{{\text{P}}_{2} }}^{{{\text{re}}}} \delta_{{l{\text{P}}_{2} }}^{{{\text{re}}}} k_{{{\text{P}}_{2} }}^{{{\text{re}}}} ,f_{53} = n_{{{\text{P}}_{3} }}^{{{\text{re}}}} \delta_{{l{\text{P}}_{3} }}^{{{\text{re}}}} k_{{{\text{P}}_{3} }}^{{{\text{re}}}} ,f_{54} = f_{55} = f_{56} = 0,f_{57} = l_{{\text{S}}}^{{{\text{re}}}} \delta_{l}^{{{\text{re}}}} k_{{\text{S}}}^{{{\text{re}}}} ,f_{58} = 0,$$
$$f_{61} = n_{{{\text{P}}_{1} }}^{{{\text{re}}}} \delta_{{g{\text{P}}_{1} }}^{{{\text{re}}}} k_{{{\text{P}}_{1} }}^{{{\text{re}}}} ,f_{62} = n_{{{\text{P}}_{2} }}^{{{\text{re}}}} \delta_{{g{\text{P}}_{2} }}^{{{\text{re}}}} k_{{{\text{P}}_{2} }}^{{{\text{re}}}} ,f_{63} = n_{{{\text{P}}_{3} }}^{{{\text{re}}}} \delta_{{g{\text{P}}_{3} }}^{{{\text{re}}}} k_{{{\text{P}}_{3} }}^{{{\text{re}}}} ,f_{64} = f_{65} = f_{66} = 0,f_{67} = l_{{\text{S}}}^{{{\text{re}}}} \delta_{g}^{{{\text{re}}}} k_{{\text{S}}}^{{{\text{re}}}} ,f_{68} = 0,$$
$$f_{71} = f_{72} = f_{73} = 0,f_{74} = - n_{{{\text{P}}_{1} }}^{{{\text{tr}}}} \delta_{{l{\text{P}}_{1} }}^{{{\text{tr}}}} k_{{{\text{P}}_{1} }}^{{{\text{tr}}}} ,f_{75} = - n_{{{\text{P}}_{2} }}^{{{\text{tr}}}} \delta_{{l{\text{P}}_{2} }}^{{{\text{tr}}}} k_{{{\text{P}}_{2} }}^{{{\text{tr}}}} f_{76} = - n_{{{\text{P}}_{3} }}^{{{\text{tr}}}} \delta_{{l{\text{P}}_{3} }}^{{{\text{tr}}}} k_{{{\text{P}}_{3} }}^{{{\text{tr}}}} ,f_{77} = 0,f_{78} = l_{{\text{S}}}^{{{\text{tr}}}} \delta_{l}^{{{\text{tr}}}} k_{{\text{S}}}^{{{\text{tr}}}} ,$$
$$f_{81} = f_{82} = f_{83} = 0,f_{84} = - n_{{{\text{P}}_{1} }}^{{{\text{tr}}}} \delta_{{g{\text{P}}_{1} }}^{{{\text{tr}}}} k_{{{\text{P}}_{1} }}^{{{\text{tr}}}} ,f_{85} = - n_{{{\text{P}}_{2} }}^{{{\text{tr}}}} \delta_{{g{\text{P}}_{2} }}^{{{\text{tr}}}} k_{{{\text{P}}_{2} }}^{{{\text{tr}}}} ,f_{86} = - n_{{{\text{P}}_{3} }}^{{{\text{tr}}}} \delta_{{g{\text{P}}_{3} }}^{{{\text{tr}}}} k_{{{\text{P}}_{3} }}^{{{\text{tr}}}} ,f_{87} = 0,f_{88} = l_{{\text{S}}}^{{{\text{tr}}}} \delta_{g}^{{{\text{tr}}}} k_{{\text{S}}}^{{{\text{tr}}}} ,$$
$$\left. \begin{gathered} g_{1} = \left( {\lambda_{c}^{\rm I} + 2\mu^{\rm I} n_{{{\text{P}}_{{1}} }}^{{{\text{in2}}}} + B_{1}^{\rm I} \delta_{{l{\text{P}}_{1} }}^{{{\text{in}}}} + B_{2}^{\rm I} \delta_{{{\text{P}}_{1} }}^{{{\text{in2}}}} } \right)k_{{{\text{P}}_{1} }}^{{{\text{in2}}}} g_{2} = - 2\mu^{\rm I} l_{{{\text{P}}_{1} }}^{{{\text{in}}}} n_{{{\text{P}}_{{1}} }}^{{{\text{in}}}} k_{{{\text{P}}_{1} }}^{{{\text{in2}}}} ,g_{3} = n_{{{\text{P}}_{{1}} }}^{{{\text{in}}}} k_{{{\text{P}}_{1} }}^{{{\text{in}}}} \hfill \\ g_{4} = - l_{{{\text{P}}_{1} }}^{{{\text{in}}}} k_{{{\text{P}}_{1} }}^{{{\text{in}}}} ,g_{5} = n_{{{\text{P}}_{{1}} }}^{{{\text{in}}}} \delta_{{l{\text{P}}_{1} }}^{{{\text{in}}}} k_{{{\text{P}}_{1} }}^{{{\text{in}}}} ,g_{6} = n_{{{\text{P}}_{{1}} }}^{{{\text{in}}}} \delta_{{g{\text{P}}_{1} }}^{{{\text{in}}}} k_{{{\text{P}}_{1} }}^{{{\text{in}}}} ,g_{7} = g_{8} = 0 \hfill \\ \end{gathered} \right\}.$$
Appendix 4
The expression of each element in Eq. (19)
$$\left. \begin{gathered} \sigma_{33}^{{rP_{1} }} = - \left( {\lambda {}_{c}^{{\text{I}}} + 2\mu^{{\text{I}}} n_{{P_{1} }}^{re2} + B_{1}^{{\text{I}}} \delta_{{lP_{1} }}^{re} + B_{2}^{{\text{I}}} \delta_{{gP_{1} }}^{re} } \right)k_{{P_{1} }}^{re2} A_{{sP_{1} }}^{re} ,\sigma_{33}^{{rP_{2} }} = - \left( {\lambda {}_{c}^{{\text{I}}} + 2\mu^{{\text{I}}} n_{{P_{2} }}^{re2} + B_{1}^{{\text{I}}} \delta_{{lP_{2} }}^{re} + B_{2}^{{\text{I}}} \delta_{{gP_{2} }}^{re} } \right)k_{{P_{2} }}^{re2} A_{{sP_{2} }}^{re} \hfill \\ \sigma_{33}^{{rP_{3} }} = - \left( {\lambda {}_{c}^{{\text{I}}} + 2\mu^{{\text{I}}} n_{{P_{3} }}^{re2} + B_{1}^{{\text{I}}} \delta_{{lP_{3} }}^{re} + B_{2}^{{\text{I}}} \delta_{{gP_{3} }}^{re} } \right)k_{{P_{3} }}^{re2} A_{{sP_{3} }}^{re} ,\sigma_{33}^{iS} = 2\mu^{{\text{I}}} l_{S}^{in} n_{S}^{in} k_{S}^{in2} {\mathbf{B}}_{s}^{in} ,\sigma_{33}^{rS} = - 2\mu^{{\text{I}}} l_{S}^{re} n_{S}^{re} k_{S}^{re2} {\mathbf{B}}_{s}^{re} \hfill \\ \end{gathered} \right\},$$
$$\left. \begin{gathered} \sigma_{13}^{{{\text{I}}\left( {iS} \right)}} = \mu^{{\text{I}}} \left( {n_{S}^{in2} - l_{S}^{in2} } \right)k_{S}^{in2} B_{s}^{in} ,\sigma_{13}^{{{\text{I}}\left( {rP_{1} } \right)}} = - 2\mu^{{\text{I}}} l_{{P_{1} }}^{re} n_{{P_{1} }}^{re} k_{{P_{1} }}^{re2} A_{{sP_{1} }}^{re} ,\sigma_{13}^{{{\text{I}}\left( {rP_{2} } \right)}} = - 2\mu^{{\text{I}}} l_{{P_{2} }}^{re} n_{{P_{2} }}^{re} k_{{P_{2} }}^{re2} A_{{sP_{2} }}^{re} \hfill \\ \sigma_{13}^{{{\text{I}}\left( {rP_{3} } \right)}} = - 2\mu^{{\text{I}}} l_{{P_{3} }}^{re} n_{{P_{3} }}^{re} k_{{P_{3} }}^{re2} A_{{sP_{3} }}^{re} \sigma_{13}^{{{\rm I}\left( {rS} \right)}} = \mu^{{\text{I}}} \left( {n_{S}^{re2} - l_{S}^{re2} } \right)k_{S}^{re2} {\mathbf{B}}_{s}^{re} \hfill \\ \end{gathered} \right\},$$
$$\left. \begin{gathered} p_{l}^{{rP_{1} }} = \left( {a_{11} + a_{12} \delta_{{lP_{1} }}^{re} + a_{13} \delta_{{gP_{1} }}^{re} } \right)k_{{P_{1} }}^{re2} A_{{sP_{1} }}^{re} ,p_{l}^{{rP_{2} }} = \left( {a_{11} + a_{12} \delta_{{lP_{2} }}^{re} + a_{13} \delta_{{gP_{2} }}^{re} } \right)k_{{P_{2} }}^{re2} A_{{sP_{2} }}^{re} ,p_{l}^{iS} = 0 \hfill \\ p_{l}^{{rP_{3} }} = \left( {a_{11} + a_{12} \delta_{{lP_{3} }}^{re} + a_{13} \delta_{{gP_{3} }}^{re} } \right)k_{{P_{3} }}^{re2} A_{{sP_{3} }}^{re} ,p_{l}^{rS} = 0 \hfill \\ \end{gathered} \right\},$$
$$\left. \begin{gathered} p_{g}^{{rP_{1} }} = \left( {a_{21} + a_{22} \delta_{{lP_{1} }}^{re} + a_{23} \delta_{{gP_{1} }}^{re} } \right)k_{{P_{1} }}^{re2} A_{{sP_{1} }}^{re} ,p_{g}^{{rP_{2} }} = \left( {a_{21} + a_{22} \delta_{{lP_{2} }}^{re} + a_{23} \delta_{{gP_{2} }}^{re} } \right)k_{{P_{2} }}^{re2} A_{{sP_{2} }}^{re} ,p_{g}^{iS} = 0 \hfill \\ p_{g}^{{rP_{3} }} = \left( {a_{21} + a_{22} \delta_{{lP_{3} }}^{re} + a_{23} \delta_{{gP_{3} }}^{re} } \right)k_{{P_{3} }}^{re2} A_{{sP_{3} }}^{re} ,p_{g}^{rS} = 0 \hfill \\ \end{gathered} \right\},$$
$$\dot{u}_{3s}^{iS} = l_{S}^{in} v_{S}^{in} k_{S}^{in2} {\mathbf{B}}_{s}^{in} ,\dot{u}_{3s}^{{rP_{1} }} = n_{{P_{1} }}^{re} v_{{P_{1} }}^{re} k_{{P_{1} }}^{re2} A_{{sP_{1} }}^{re} ,\dot{u}_{3s}^{{rP_{2} }} = n_{{P_{2} }}^{re} v_{{P_{2} }}^{re} k_{{P_{2} }}^{re2} A_{{sP_{2} }}^{re} ,\dot{u}_{3s}^{{rP_{3} }} = n_{{P_{3} }}^{re} v_{{P_{3} }}^{re} k_{{P_{3} }}^{re2} A_{{sP_{3} }}^{re} ,\dot{u}_{3s}^{rS} = l_{S}^{re} v_{S}^{re} k_{S}^{re2} {\mathbf{B}}_{s}^{re} ,$$
$$\dot{u}_{1s}^{iS} = n_{S}^{in} v_{S}^{in} k_{S}^{in2} {\mathbf{B}}_{s}^{in} ,\dot{u}_{1s}^{{rP_{1} }} = l_{{P_{1} }}^{re} v_{{P_{1} }}^{re} k_{{P_{1} }}^{re2} A_{{sP_{1} }}^{re} ,\dot{u}_{1s}^{{rP_{2} }} = l_{{P_{2} }}^{re} v_{{P_{2} }}^{re} k_{{P_{2} }}^{re2} A_{{sP_{2} }}^{re} ,\dot{u}_{1s}^{{rP_{3} }} = l_{{P_{3} }}^{re} v_{{P_{3} }}^{re} k_{{P_{3} }}^{re2} A_{{sP_{3} }}^{re} ,\dot{u}_{1s}^{rS} = - n_{S}^{re} v_{S}^{re} k_{S}^{re2} {\mathbf{B}}_{s}^{re} ,$$
$$\left. \begin{gathered} \dot{u}_{3l}^{iS} = l_{S}^{in} v_{S}^{in} k_{S}^{in2} \delta_{ls}^{re} {\mathbf{B}}_{s}^{in} ,\dot{u}_{3l}^{{rP_{1} }} = n_{{P_{1} }}^{re} v_{{P_{1} }}^{re} k_{{P_{1} }}^{re2} \delta_{{lP_{1} }}^{re} A_{{sP_{1} }}^{re} ,\dot{u}_{3l}^{{rP_{2} }} = n_{{P_{2} }}^{re} v_{{P_{2} }}^{re} k_{{P_{2} }}^{re2} \delta_{{lP_{2} }}^{re} A_{{sP_{2} }}^{re} ,\dot{u}_{3l}^{{rP_{3} }} = n_{{P_{3} }}^{re} v_{{P_{3} }}^{re} k_{{P_{3} }}^{re2} \delta_{{lP_{3} }}^{re} A_{{sP_{3} }}^{re} \hfill \\ \dot{u}_{3l}^{rS} = l_{S}^{re} v_{S}^{re} k_{S}^{re2} \delta_{ls}^{re} {\mathbf{B}}_{s}^{re} \hfill \\ \end{gathered} \right\},$$
$$\left. \begin{gathered} \dot{u}_{3g}^{iS} = l_{S}^{in} v_{S}^{in} k_{S}^{in2} \delta_{gs}^{re} {\mathbf{B}}_{s}^{in} ,\dot{u}_{3g}^{{rP_{1} }} = n_{{P_{1} }}^{re} v_{{P_{1} }}^{re2} k_{{P_{1} }}^{2} \delta_{{gP_{1} }}^{re} A_{{sP_{1} }}^{re} ,\dot{u}_{3g}^{{rP_{2} }} = n_{{P_{2} }}^{re} v_{{P_{2} }}^{re} k_{{P_{2} }}^{re2} \delta_{{gP_{2} }}^{re} A_{{sP_{2} }}^{re} ,\dot{u}_{3g}^{{rP_{3} }} = n_{{P_{3} }}^{re} v_{{P_{3} }}^{re} k_{{P_{3} }}^{re2} \delta_{{lP_{3} }}^{re} A_{{gP_{3} }}^{re} \hfill \\ \dot{u}_{3g}^{rS} = l_{S}^{re} v_{S}^{re} k_{S}^{re2} \delta_{gs}^{re} {\mathbf{B}}_{s}^{re} \hfill \\ \end{gathered} \right\}.$$
Appendix 5
The expression of each element in Eq. (22)
$$\left. \begin{gathered} \sigma_{33}^{{{\text{II}}\left( {tP_{1} } \right)}} = - \left( {\lambda_{c}^{{{\text{II}}}} + 2\mu^{{{\text{II}}}} n_{{P_{1} }}^{tr2} + B_{{_{1} }}^{{{\text{II}}}} \delta_{{lP_{1} }}^{tr} + B_{2}^{{{\text{II}}}} \delta_{{gP_{1} }}^{tr} } \right)k_{{P_{1} }}^{tr2} A_{{sP_{1} }}^{tr} ,\sigma_{33}^{{{\text{II}}\left( {tP_{2} } \right)}} = - \left( {\lambda_{c}^{{{\text{II}}}} + 2\mu^{{{\text{II}}}} n_{{P_{2} }}^{tr2} + B_{{_{1} }}^{{{\text{II}}}} \delta_{{lP_{2} }}^{tr} + B_{2}^{{{\text{II}}}} \delta_{{gP_{2} }}^{tr} } \right)k_{{P_{2} }}^{tr2} A_{{sP_{2} }}^{tr} \hfill \\ \sigma_{33}^{{{\text{II}}\left( {tP_{3} } \right)}} = - \left( {\lambda_{c}^{{{\text{II}}}} + 2\mu^{{{\text{II}}}} n_{{P_{3} }}^{tr2} + B_{{_{1} }}^{{{\text{II}}}} \delta_{{lP_{3} }}^{tr} + B_{2}^{{{\text{II}}}} \delta_{{gP_{3} }}^{tr} } \right)k_{{P_{3} }}^{tr2} A_{{sP_{3} }}^{tr} ,\sigma_{33}^{{{\text{II}}\left( {tS} \right)}} = 2\mu^{{{\text{II}}}} l_{S}^{tr} n_{S}^{tr} k_{S}^{tr2} {\mathbf{B}}_{s}^{tr} \hfill \\ \end{gathered} \right\},$$
$$\sigma_{13}^{{{\text{II}}\left( {tP_{1} } \right)}} = 2\mu^{{{\text{II}}}} l_{{P_{1} }}^{tr} n_{{P_{1} }}^{tr} k_{{P_{1} }}^{tr2} A_{{sP_{1} }}^{tr} ,\sigma_{13}^{{{\text{II}}\left( {tP_{2} } \right)}} = 2\mu^{{{\text{II}}}} l_{{P_{2} }}^{tr} n_{{P_{2} }}^{tr} k_{{P_{2} }}^{tr2} A_{{sP_{2} }}^{tr} \sigma_{13}^{{{\text{II}}\left( {tP_{3} } \right)}} = 2\mu^{{{\text{II}}}} l_{{P_{3} }}^{tr} n_{{P_{3} }}^{tr} k_{{P_{3} }}^{tr2} A_{{sP_{3} }}^{tr} ,\sigma_{13}^{{{\text{II}}\left( {tS} \right)}} = \mu^{{{\text{II}}}} \left( {n_{S}^{tr2} - l_{S}^{tr2} } \right)k_{S}^{tr2} {\mathbf{B}}_{s}^{tr} ,$$
$$\left. \begin{gathered} p_{l}^{{tP_{1} }} = \left( {a_{11} + a_{12} \delta_{{lP_{1} }}^{tr} + a_{13} \delta_{{gP_{1} }}^{tr} } \right)k_{{P_{1} }}^{tr2} A_{{sP_{1} }}^{tr} ,p_{l}^{{tP_{2} }} = \left( {a_{11} + a_{12} \delta_{{lP_{2} }}^{tr} + a_{13} \delta_{{gP_{2} }}^{tr} } \right)k_{{P_{2} }}^{tr2} A_{{sP_{2} }}^{tr} \hfill \\ p_{l}^{{tP_{3} }} = \left( {a_{11} + a_{12} \delta_{{lP_{3} }}^{tr} + a_{13} \delta_{{gP_{3} }}^{tr} } \right)k_{{P_{3} }}^{tr2} A_{{sP_{3} }}^{tr} ,p_{l}^{tS} = 0 \hfill \\ \end{gathered} \right\},$$
$$\left. \begin{gathered} p_{g}^{{tP_{1} }} = \left( {a_{21} + a_{22} \delta_{{lP_{1} }}^{tr} + a_{23} \delta_{{gP_{1} }}^{tr} } \right)k_{{P_{1} }}^{tr2} A_{{sP_{1} }}^{tr} ,p_{g}^{{tP_{2} }} = \left( {a_{21} + a_{22} \delta_{{lP_{2} }}^{tr} + a_{23} \delta_{{gP_{2} }}^{tr} } \right)k_{{P_{2} }}^{tr2} A_{{sP_{2} }}^{tr} \hfill \\ p_{g}^{{tP_{3} }} = \left( {a_{21} + a_{22} \delta_{{lP_{3} }}^{tr} + a_{23} \delta_{{gP_{3} }}^{tr} } \right)k_{{P_{3} }}^{tr2} A_{{sP_{3} }}^{tr} ,p_{g}^{tS} = 0 \hfill \\ \end{gathered} \right\},$$
$$\dot{u}_{1s}^{{tP_{1} }} = l_{{P_{1} }}^{tr} v_{{P_{1} }}^{tr} k_{{P_{1} }}^{tr2} A_{{sP_{1} }}^{tr} ,\dot{u}_{1s}^{{tP_{2} }} = l_{{P_{2} }}^{tr} v_{{P_{2} }}^{tr} k_{{P_{2} }}^{tr2} A_{{sP_{2} }}^{tr} ,\dot{u}_{1s}^{{tP_{3} }} = l_{{P_{3} }}^{tr} v_{{P_{3} }}^{tr} k_{{P_{3} }}^{tr2} A_{{sP_{3} }}^{tr} ,\dot{u}_{1s}^{tS} = n_{S}^{tr} v_{S}^{tr} k_{S}^{tr2} {\mathbf{\rm B}}_{s}^{tr} ,$$
$$\dot{u}_{3s}^{{tP_{1} }} = - n_{{P_{1} }}^{tr} v_{{P_{1} }}^{tr} k_{{P_{1} }}^{tr2} A_{{sP_{1} }}^{tr} ,\dot{u}_{3s}^{{tP_{2} }} = - n_{{P_{2} }}^{tr} v_{{P_{2} }}^{tr} k_{{P_{2} }}^{tr2} A_{{sP_{2} }}^{tr} ,\dot{u}_{3s}^{{tP_{3} }} = - n_{{P_{3} }}^{tr} v_{{P_{3} }}^{tr} k_{{P_{3} }}^{tr2} A_{{sP_{3} }}^{tr} ,\dot{u}_{3s}^{tS} = l_{S}^{re} v_{S}^{tr} k_{S}^{tr2} {\mathbf{B}}_{s}^{tr} ,$$
$$\dot{u}_{3l}^{{tP_{1} }} = - n_{{P_{1} }}^{tr} v_{{P_{1} }}^{tr} k_{{P_{1} }}^{tr2} \delta_{{lP_{1} }}^{tr} A_{{sP_{1} }}^{tr} ,\dot{u}_{3l}^{{tP_{2} }} = - n_{{P_{2} }}^{tr} v_{{P_{2} }}^{tr} k_{{P_{2} }}^{tr2} \delta_{{lP_{2} }}^{tr} A_{{sP_{2} }}^{tr} ,\dot{u}_{3l}^{{tP_{3} }} = - n_{{P_{3} }}^{tr} v_{{P_{3} }}^{tr} k_{{P_{3} }}^{tr2} \delta_{{lP_{3} }}^{tr} A_{{sP_{3} }}^{tr} ,\dot{u}_{3l}^{tS} = l_{S}^{tr} v_{S}^{tr} k_{S}^{tr2} \delta_{ls}^{tr} {\mathbf{B}}_{s}^{tr} ,$$
$$\dot{u}_{3g}^{{tP_{1} }} = - n_{{P_{1} }}^{tr} v_{{P_{1} }}^{tr} k_{{P_{1} }}^{tr2} \delta_{{gP_{1} }}^{tr} A_{{sP_{1} }}^{tr} ,\dot{u}_{3g}^{{tP_{2} }} = - n_{{P_{2} }}^{tr} v_{{P_{2} }}^{tr} k_{{P_{2} }}^{tr2} \delta_{{gP_{2} }}^{tr} A_{{sP_{2} }}^{tr} ,\dot{u}_{3g}^{{tP_{3} }} = - n_{{P_{3} }}^{tr} v_{{P_{3} }}^{tr} k_{{P_{3} }}^{tr2} \delta_{{gP_{3} }}^{tr} A_{{sP_{3} }}^{tr} ,\dot{u}_{3g}^{tS} = l_{S}^{tr} v_{S}^{tr} k_{S}^{tr2} \delta_{gs}^{tr} {\mathbf{B}}_{s}^{tr} .$$
