Appendix A: The specific form of coefficient in Eq. (1)
The specific form of coefficient in Eqs. (1a) − (1d), \({B}_{1}\), \({B}_{2}\), \({B}_{3}\), \({B}_{4}\), \({B}_{5}\), \({B}_{6}\), \({B}_{7}\), \({B}_{8}\), \({C}_{1}\), \({C}_{2}\), \({C}_{3}\), \({C}_{4}\), \({D}_{0}\), \({D}_{1}\), \({D}_{2}\), \({D}_{3}\), \({\vartheta }^{l}\), \({\vartheta }^{g}\), \({\upsilon }^{l}\) and \({\upsilon }^{g}\) is given as follows (Zhou et al. [27]):
$$\left.\begin{array}{c}{B}_{1}=\frac{{A}_{22}}{{A}_{22}{A}_{11}-{A}_{12}{A}_{21}}, {B}_{2}=\frac{{A}_{22}{A}_{14}-{A}_{12}{A}_{24}}{{A}_{14}{(A}_{22}{A}_{11}-{A}_{12}{A}_{21})}, {B}_{3}=\frac{{A}_{22}{A}_{15}-{A}_{12}{A}_{25}}{{A}_{15}{(A}_{22}{A}_{11}-{A}_{12}{A}_{21})}, {B}_{4}=\frac{{A}_{22}{A}_{16}-{A}_{12}{A}_{26}}{{A}_{22}{A}_{11}-{A}_{12}{A}_{21}}\\ {B}_{5}=\frac{{A}_{21}}{{A}_{21}{A}_{12}-{A}_{11}{A}_{22}}, {B}_{6}=\frac{{A}_{21}{A}_{14}-{A}_{11}{A}_{24}}{{A}_{14}{(A}_{21}{A}_{12}-{A}_{11}{A}_{22})}, {B}_{7}=\frac{{A}_{21}{A}_{15}-{A}_{11}{A}_{25}}{{A}_{15}{(A}_{21}{A}_{12}-{A}_{11}{A}_{22})}, {B}_{8}=\frac{{A}_{21}{A}_{16}-{A}_{11}{A}_{26}}{{A}_{21}{A}_{12}-{A}_{11}{A}_{22}}\end{array}\right\}$$
(A.1)
$$\left.\begin{array}{c}{C}_{1}={\lambda }^{^{\prime}}{T}_{0}+{\beta }_{T}{T}_{0}[{B}_{1}\gamma +{B}_{5}(1-\gamma )], {C}_{2}={\beta }_{T}{T}_{0}[{B}_{2}\gamma +{B}_{6}(1-\gamma )]\\ {C}_{3}={\beta }_{T}{T}_{0}\left[{B}_{3}\gamma +{B}_{7}\left(1-\gamma \right)\right],{C}_{4}=\tilde{m }+{\beta }_{T}{T}_{0}\left[{B}_{4}\gamma +{B}_{8}\left(1-\gamma \right)\right]\end{array}\right\}$$
(A.2)
$$\left.\begin{array}{c}{D}_{0}=\alpha \gamma {B}_{1}+\alpha \left(1-\gamma \right){B}_{5}, {D}_{1}=\alpha \gamma {B}_{2}+\alpha \left(1-\gamma \right){B}_{6}\\ {D}_{2}=\alpha \gamma {B}_{3}+\alpha \left(1-\gamma \right){B}_{7}, {D}_{3}=\alpha \gamma {B}_{4}+\alpha \left(1-\gamma \right){B}_{8}-{\lambda }^{^{\prime}}\end{array}\right\}$$
(A.3)
$${\vartheta }^{l}=\frac{{\rho }^{l}}{n{S}^{l}}, {\vartheta }^{g}=\frac{{\rho }^{g}}{n\left(1-{S}^{l}\right)}, {\upsilon }^{l}=\frac{{\mu }_{l}}{{k}_{r}^{l}k}, {\upsilon }^{g}=\frac{{\mu }_{g}}{{k}_{r}^{g}k}$$
(A.4)
with
$$\left.\begin{array}{c}{A}_{11}=n{S}^{l}{\beta }_{wp}, {A}_{12}=n\left(1-{S}^{l}\right)\frac{{M}_{a}}{{\rho }^{g}RT}, {A}_{13}=1-n \\ {A}_{14}=n{S}^{l}, {A}_{15}=n(1-{S}^{l}), {A}_{16}=-\left(1-n\right){\beta }_{sT}-{nS}^{l}{\beta }_{wT}-n(1-{S}^{l})\frac{{M}_{a}{p}_{g}}{{\rho }^{g}R{T}^{2}}\end{array}\right\}$$
(A.5)
$$\left.\begin{array}{c}{A}_{21}=n{S}^{l}\left(1-{S}^{l}\right){\beta }_{wp}-n{A}_{s}, {A}_{22}=n{A}_{s}-n{S}^{l}\left(1-{S}^{l}\right)\frac{{M}_{a}}{{\rho }^{g}RT}, {A}_{23}=0 \\ {A}_{24}=-{A}_{25}=n{S}^{l}\left(1-{S}^{l}\right), {A}_{26}=-n{\beta }_{\psi }{A}_{s}{\chi }^{-1}{\left({S}_{e}^{-1/m}-1\right)}^{1/d}+ n{S}^{l}(1-{S}^{l})(\frac{{M}_{a}{p}_{g}}{{\rho }^{g}R{T}^{2}}-{\beta }_{wT})\end{array}\right\}$$
(A.6)
$${\lambda }^{^{\prime}}=3{K}_{b}{\beta }_{T}, \tilde{m }=\left(1-n\right){\rho }^{s}{c}_{s}+ n{S}^{l}{\rho }^{l}{c}_{l}+n(1-{S}^{l}){\rho }^{g}{c}_{g}, \alpha =1-{K}_{b}/{K}_{s}$$
(A.7)
$$\left.\begin{array}{c}{\mu }_{l}=\left(243.18\times {10}^{-7}\right){10}^{\frac{247.8}{T-140}}, {\mu }_{g}=1.48\times {10}^{-6}\frac{\sqrt{T}}{1+119/T} \\ {k}_{r}^{l}=\sqrt{{S}_{e}}{\left[1-{\left(1-{S}_{e}^{1/m}\right)}^{m}\right]}^{2},{k}_{r}^{g}=\sqrt{1-{S}_{e}}{\left(1-{S}_{e}^{1/m}\right)}^{2m}\end{array}\right\}$$
(A.8)
in which \(\gamma \) denotes the effective stress parameter, and \(\gamma \) is identical to the saturation \({S}^{l}\) for approximation. \(m\), \(d\) and \(\chi \) are the material parameters of V–G model. The two set of parameters, \({\beta }_{sp}\), \({\beta }_{wp}\) and \({\beta }_{sT}\), \({\beta }_{wT}\), designate the compressibility and the thermal expansion coefficients of solid particle and liquid, respectively. The symbols \({\beta }_{\psi }\) and \({\beta }_{T}\) denote the surface tension-temperature dependent coefficient and the thermal expansion coefficient of medium, respectively. \({c}_{s}\), \({c}_{l}\) and \({c}_{g}\) represent the specific heat capacity of solid, liquid and gas phases, respectively. \({K}_{s}\) denotes the compressibility moduli of the solid grains. \({K}_{b}=\lambda +2\mu /3\) denotes the bulk modulus of soil skeleton. \(k\) is the intrinsic permeability. \({M}_{a}=0.0288 \mathrm{kg}/\mathrm{mol}\) is the molar mass of dry air. \(R=8.3144 \mathrm{J}/\mathrm{mol}/\mathrm{K}\) is the universal gas constant. \({T}_{0}\) denotes reference temperature. The effective water saturation \({S}_{e}\) is defined as \({S}_{e}=\left({S}^{l}-{S}_{res}^{l}\right)/\left({S}_{sat}^{l}-{S}_{res}^{l}\right)\), and the symbols \({S}_{res}^{l}\) and \({S}_{sat}^{l}\) stand for the degree of saturation at the residual state and at full saturation, respectively. The symbol \({A}_{s}\) can be expressed as \({A}_{s}=-\chi md\left({S}_{sat}^{l}-{S}_{res}^{l}\right){S}_{e}^{(m+1)/m}{\left({S}_{e}^{-1/m}-1\right)}^{(d-1)/d}\).
Appendix B: The terms in Eqs. (5a) and (5b)
The terms in Eqs. (5a) and (5b) in the text are given as follows:
$${b}_{11}=\rho {\omega }^{2}-(\overline{\lambda }+2\mu ){k}_{p}^{2}, {b}_{12}={\rho }^{l}{\omega }^{2}-{D}_{1}{k}_{p}^{2}, {b}_{13}={\rho }^{g}{\omega }^{2}-{D}_{2}{k}_{p}^{2}, {b}_{14}={D}_{3}$$
(B.1)
$${b}_{21}={\rho }^{l}{\omega }^{2}-{B}_{1}{k}_{p}^{2}, {b}_{22}={\vartheta }^{l}{\omega }^{2}+{\upsilon }^{l}\mathrm{i}\omega -{B}_{2}{k}_{p}^{2}, {b}_{23}=-{B}_{3}{k}_{p}^{2}, {b}_{24}={B}_{4}$$
(B.2)
$${b}_{31}={\rho }^{g}{\omega }^{2}-{B}_{5}{k}_{p}^{2}, {b}_{32}=-{B}_{6}{k}_{p}^{2}, {b}_{33}={\vartheta }^{g}{\omega }^{2}+{\upsilon }^{g}\mathrm{i}\omega -{B}_{7}{k}_{p}^{2}, {b}_{34}={B}_{8}$$
(B.3)
$${b}_{41}={C}_{1}(\mathrm{i}\omega +{\tau }_{q}{\omega }^{2}){k}_{p}^{2}, {b}_{42}={C}_{2}(\mathrm{i}\omega +{\tau }_{q}{\omega }^{2}){k}_{p}^{2}$$
(B.4)
$${b}_{43}={C}_{3}(\mathrm{i}\omega +{\tau }_{q}{\omega }^{2}){k}_{p}^{2}, {b}_{44}={{K(1-{\tau }_{\theta }\mathrm{i}\omega )k}_{p}^{2}-C}_{4}(\mathrm{i}\omega +{\tau }_{q}{\omega }^{2})$$
(B.5)
$$\left.\begin{array}{l}{c}_{11}=\rho {\omega }^{2}-\mu {k}_{s}^{2}, {c}_{12}={\rho }^{l}{\omega }^{2}, {c}_{13}={\rho }^{g}{\omega }^{2}, {c}_{21}={\rho }^{l}{\omega }^{2} \\ {c}_{22}={\vartheta }^{l}{\omega }^{2}+{\upsilon }^{l}i\omega , {c}_{31}={\rho }^{g}{\omega }^{2}, {c}_{33}={\vartheta }^{g}{\omega }^{2}+{\upsilon }^{g}i\omega , {c}_{23}={c}_{32}=0\end{array}\right\}$$
(B.6)
Appendix C: The terms in Eqs. (6a)–(6c)
The terms \({d}_{11}^{(\tilde{n })}\sim {d}_{36}^{(\tilde{n })}\) in Eqs. (6a) − (6c) in the text are given as follows:
$$\left.\begin{array}{c}{d}_{11}^{(\tilde{n })}={b}_{31}^{(\tilde{n })}{b}_{43}^{(\tilde{n })}-{b}_{41}^{\left(\tilde{n }\right)}{b}_{33}^{\left(\tilde{n }\right)}, {d}_{12}^{\left(\tilde{n }\right)}={b}_{11}^{\left(\tilde{n }\right)}{b}_{23}^{\left(\tilde{n }\right)}-{b}_{21}^{\left(\tilde{n }\right)}{b}_{13}^{\left(\tilde{n }\right)}, {d}_{13}^{(\tilde{n })}={b}_{12}^{(\tilde{n })}{b}_{23}^{(\tilde{n })}-{b}_{22}^{(\tilde{n })}{b}_{13}^{(\tilde{n })}\\ {d}_{14}^{(\tilde{n })}={b}_{32}^{(\tilde{n })}{b}_{43}^{(\tilde{n })}-{b}_{42}^{\left(\tilde{n }\right)}{b}_{33}^{\left(\tilde{n }\right)}, {d}_{15}^{\left(\tilde{n }\right)}={b}_{14}^{\left(\tilde{n }\right)}{b}_{23}^{\left(\tilde{n }\right)}-{b}_{24}^{\left(\tilde{n }\right)}{b}_{13}^{\left(\tilde{n }\right)},{d}_{16}^{(\tilde{n })}={b}_{34}^{(\tilde{n })}{b}_{43}^{(\tilde{n })}-{b}_{44}^{(\tilde{n })}{b}_{33}^{(\tilde{n })}\end{array}\right\}$$
(C.1)
$$\left.\begin{array}{c}{d}_{21}^{\left(\tilde{n }\right)}={b}_{31}^{\left(\tilde{n }\right)}{b}_{42}^{\left(\tilde{n }\right)}-{b}_{41}^{\left(\tilde{n }\right)}{b}_{32}^{\left(\tilde{n }\right)}, {d}_{22}^{\left(\tilde{n }\right)}={b}_{11}^{\left(\tilde{n }\right)}{b}_{22}^{\left(\tilde{n }\right)}-{b}_{21}^{\left(\tilde{n }\right)}{b}_{12}^{\left(\tilde{n }\right)}, {d}_{23}^{\left(\tilde{n }\right)}={b}_{13}^{\left(\tilde{n }\right)}{b}_{22}^{\left(\tilde{n }\right)}-{b}_{23}^{\left(\tilde{n }\right)}{b}_{12}^{\left(\tilde{n }\right)}\\ {d}_{24}^{(\tilde{n })}={b}_{33}^{(\tilde{n })}{b}_{42}^{(\tilde{n })}-{b}_{43}^{\left(\tilde{n }\right)}{b}_{32}^{\left(\tilde{n }\right)},{d}_{25}^{\left(\tilde{n }\right)}={b}_{14}^{\left(\tilde{n }\right)}{b}_{22}^{\left(\tilde{n }\right)}-{b}_{24}^{\left(\tilde{n }\right)}{b}_{12}^{\left(\tilde{n }\right)} ,{d}_{26}^{(\tilde{n })}={b}_{34}^{(\tilde{n })}{b}_{42}^{(\tilde{n })}-{b}_{44}^{(\tilde{n })}{b}_{32}^{(\tilde{n })}\end{array}\right\}$$
(C.2)
$$\left.\begin{array}{c}{d}_{31}^{\left(\tilde{n }\right)}={b}_{31}^{\left(\tilde{n }\right)}{b}_{43}^{\left(\tilde{n }\right)}-{b}_{41}^{\left(\tilde{n }\right)}{b}_{33}^{\left(\tilde{n }\right)}, {d}_{32}^{\left(\tilde{n }\right)}={b}_{11}^{\left(\tilde{n }\right)}{b}_{23}^{\left(\tilde{n }\right)}-{b}_{21}^{\left(\tilde{n }\right)}{b}_{13}^{\left(\tilde{n }\right)}, {d}_{33}^{\left(\tilde{n }\right)}={b}_{14}^{\left(\tilde{n }\right)}{b}_{23}^{\left(\tilde{n }\right)}-{b}_{24}^{\left(\tilde{n }\right)}{b}_{13}^{\left(\tilde{n }\right)}\\ {d}_{34}^{(\tilde{n })}={b}_{34}^{(\tilde{n })}{b}_{43}^{(\tilde{n })}-{b}_{44}^{\left(\tilde{n }\right)}{b}_{33}^{\left(\tilde{n }\right)}{, d}_{35}^{\left(\tilde{n }\right)}={b}_{12}^{\left(\tilde{n }\right)}{b}_{23}^{\left(\tilde{n }\right)}-{b}_{22}^{\left(\tilde{n }\right)}{b}_{13}^{\left(\tilde{n }\right)}, {d}_{36}^{(\tilde{n })}={b}_{32}^{(\tilde{n })}{b}_{43}^{(\tilde{n })}-{b}_{42}^{(\tilde{n })}{b}_{33}^{(\tilde{n })}\end{array}\right\}$$
(C.3)
Appendix D: The terms in Eq. (12a)
The expressions of terms \({k}_{11}\), \({k}_{12}\), \({k}_{21}\) and \({k}_{22}\) in Eq. (12a) in the text are given as follows:
$$\left.\begin{array}{l}{k}_{11}={\rho }^{s}{\omega }^{2}-\left(\lambda +2\mu \right){k}_{p}^{2}, {k}_{12}=-3{K}_{b}{\beta }_{T} \\ {k}_{21}=3{K}_{b}{\beta }_{T}{T}_{0}(i\omega +{\tau }_{q}{\omega }^{2}){k}_{p}^{2}, {k}_{22}=K\left(1-{\tau }_{\theta }\mathrm{i}\omega \right){k}_{p}^{2}-{\rho }^{s}{c}_{s}(i\omega +{\tau }_{q}{\omega }^{2})\end{array}\right\}$$
(D.1)
Appendix E: The elements in the matrixes [F] and [G] of Eq. (17)
The elements \({f}_{11}\sim {f}_{88}\) in the matrix [F] of Eq. (17) and the elements \({g}_{1}\sim {g}_{8}\) in the matrix [G] of Eq. (17) are given as follows:
$$\left.\begin{array}{l}{f}_{11}=\left(\overline{\lambda }+2\mu {n}_{tp1}^{2}+{D}_{1}{\delta }_{fp1}+{D}_{2}{\delta }_{ap1}\right){k}_{tp1}^{2}-{D}_{3}{\delta }_{Tp1} \\ {f}_{12}=\left(\overline{\lambda }+2\mu {n}_{tp2}^{2}+{D}_{1}{\delta }_{fp2}+{D}_{2}{\delta }_{ap2}\right){k}_{tp2}^{2}-{D}_{3}{\delta }_{Tp2} \\ {f}_{13}=\left(\overline{\lambda }+2\mu {n}_{tp3}^{2}+{D}_{1}{\delta }_{fp3}+{D}_{2}{\delta }_{ap3}\right){k}_{tp3}^{2}-{D}_{3}{\delta }_{Tp3} \\ {f}_{14}=\left(\overline{\lambda }+2\mu {n}_{tp4}^{2}+{D}_{1}{\delta }_{fp4}+{D}_{2}{\delta }_{ap4}\right){k}_{tp4}^{2}-{D}_{3}{\delta }_{Tp4} \\ {f}_{15}=-2\mu {l}_{ts}{n}_{ts}{k}_{ts}^{2}, {f}_{16}=-\left(\widehat{\lambda }+2\widehat{\mu }{n}_{rp1}^{2}\right){k}_{rp1}^{2}-3{\widehat{K}}_{b}{{\widehat{\beta }}_{T}\widehat{\delta }}_{Tp1} \\ {f}_{17}=-\left(\widehat{\lambda }+2\widehat{\mu }{n}_{rp2}^{2}\right){k}_{rp2}^{2}-3{\widehat{K}}_{b}{{\widehat{\beta }}_{T}\widehat{\delta }}_{Tp2} ,{f}_{18}=-2\widehat{\mu }{l}_{rs}{n}_{rs}{k}_{rs}^{2}\end{array}\right\}$$
(E.1)
$$\left.\begin{array}{c}{f}_{21}=2\mu {l}_{tp1}{n}_{tp1}{k}_{tp1}^{2}, {f}_{22}=2\mu {l}_{tp2}{n}_{tp2}{k}_{tp2}^{2}, {f}_{23}=2\mu {l}_{tp3}{n}_{tp3}{k}_{tp3}^{2},{f}_{24}=2\mu {l}_{tp4}{n}_{tp4}{k}_{tp4}^{2} \\ {f}_{25}=\mu {({n}_{ts}^{2}-{l}_{ts}^{2})k}_{ts}^{2}, {f}_{26}=2\widehat{\mu }{l}_{rp1}{n}_{rp1}{k}_{rp1}^{2},{f}_{27}=2\widehat{\mu }{l}_{rp2}{n}_{rp2}{k}_{rp2}^{2},{f}_{28}=\widehat{\mu }{({l}_{rs}^{2}-{n}_{rs}^{2})k}_{rs}^{2}\end{array}\right\}$$
(E.2)
$$\left.\begin{array}{c}{f}_{31}={l}_{tp1}{k}_{tp1}, {f}_{32}={l}_{tp2}{k}_{tp2}, {f}_{33}={l}_{tp3}{k}_{tp3}, {f}_{34}={l}_{tp4}{k}_{tp4} \\ {f}_{35}={n}_{ts}{k}_{ts}, {f}_{36}={-l}_{rp1}{k}_{rp1}, {f}_{37}={-l}_{rp2}{k}_{rp2},{f}_{38}={n}_{rs}{k}_{rs}\end{array}\right\}$$
(E.3)
$$\left.\begin{array}{c}{f}_{41}={n}_{tp1}{k}_{tp1}, {f}_{42}={n}_{tp2}{k}_{tp2}, {f}_{43}={n}_{tp3}{k}_{tp3}, {f}_{44}={n}_{tp4}{k}_{tp4}\\ {f}_{45}=-{l}_{ts}{k}_{ts}, {f}_{46}={n}_{rp1}{k}_{rp1}, {f}_{47}={n}_{rp2}{k}_{rp2},{f}_{48}={l}_{rs}{k}_{rs}\end{array}\right\}$$
(E.4)
$$\left.\begin{array}{c}{f}_{51}={n}_{tp1}{k}_{tp1}{\delta }_{fp1}, {f}_{52}={n}_{tp2}{k}_{tp2}{\delta }_{fp2}, {f}_{53}={n}_{tp3}{k}_{tp3}{\delta }_{fp3}\\ {f}_{54}={n}_{tp4}{k}_{tp4}{\delta }_{fp4}, {f}_{55}={-l}_{ts}{k}_{ts}{\delta }_{fs},{f}_{56}={f}_{57}={f}_{58}=0\end{array}\right\}$$
(E.5)
$$\left.\begin{array}{c}{f}_{61}={n}_{tp1}{k}_{tp1}{\delta }_{ap1}, {f}_{62}={n}_{tp2}{k}_{tp2}{\delta }_{ap2}, {f}_{63}={n}_{tp3}{k}_{tp3}{\delta }_{ap3}\\ {f}_{64}={n}_{tp4}{k}_{tp4}{\delta }_{ap4}, {f}_{65}=-{l}_{ts}{k}_{ts}{\delta }_{as}, {f}_{66}={f}_{67}={f}_{68}=0\end{array}\right\}$$
(E.6)
$$\left.\begin{array}{c}{f}_{71}={\delta }_{Tp1}, {f}_{72}={\delta }_{Tp2}, {f}_{73}={\delta }_{Tp3}, {f}_{74}={\delta }_{Tp4 } \\ {f}_{75}=0, {f}_{76}=-{\widehat{\delta }}_{Tp1}, {f}_{77}=-{\widehat{\delta }}_{Tp2}, {f}_{78}=0\end{array}\right\}$$
(E.7)
$$\left.\begin{array}{l}{f}_{81}=K{n}_{tp1}{k}_{tp1}{\delta }_{Tp1}, {f}_{82}=K{n}_{tp2}{k}_{tp2}{\delta }_{Tp2}, {f}_{83}=K{n}_{tp3}{k}_{tp3}{\delta }_{Tp3} \\ {f}_{84}=K{n}_{tp4}{k}_{tp4}{\delta }_{Tp4}, {f}_{85}={f}_{88}=0, {f}_{86}=\widehat{K}{n}_{rp1}{k}_{rp1}{\widehat{\delta }}_{Tp1},{f}_{87}=\widehat{K}{n}_{rp2}{k}_{rp2}{\widehat{\delta }}_{Tp2}\end{array}\right\}$$
(E.8)
$$\left.\begin{array}{c}{g}_{1}=\left(\widehat{\lambda }+2\widehat{\mu }{n}_{ip1}^{2}\right){k}_{ip1}^{2}+3{\widehat{K}}_{b}{{\widehat{\beta }}_{T}\widehat{\delta }}_{Tp1}, {g}_{2}=2\widehat{\mu }{l}_{ip1}{n}_{ip1}{k}_{ip1}^{2}, {g}_{3}={l}_{ip1}{k}_{ip1}\\ {g}_{4}={n}_{ip1}{k}_{ip1}, {g}_{5}={g}_{6}=0, {g}_{7}={\widehat{\delta }}_{Tp1},{g}_{8}=\widehat{K}{n}_{ip1}{k}_{ip1}{\widehat{\delta }}_{Tp1}\end{array}\right\}$$
(E.9)
Appendix F: The elements in the matrixes \({{\varvec{S}}}_{5\times 4}\), \({{\varvec{U}}}_{4\times 5}\), \({\widehat{{\varvec{S}}}}_{4\times 2}\) and \({\widehat{{\varvec{U}}}}_{2\times 4}\) of Eqs. (20a) and (20b)
The expressions of the element in \({{\varvec{S}}}_{5\times 4}\), \({{\varvec{U}}}_{4\times 5}\), \({\widehat{{\varvec{S}}}}_{4\times 2}\) and \({\widehat{{\varvec{U}}}}_{2\times 4}\) of Eqs. (20a) and (20b) in the text are given as follows:
$$\left.\begin{array}{l}{\sigma }_{zz}^{tp1}=\left[{D}_{3}{\delta }_{Tp1}-\left(\overline{\lambda }+{2\mu n}_{tp1}^{2}+{D}_{1}{\delta }_{fp1}+{D}_{2}{\delta }_{ap1}\right){k}_{tp1}^{2}\right]{A}_{tp1}^{s}\\ {\sigma }_{zz}^{tp2}=\left[{D}_{3}{\delta }_{Tp2}-\left(\overline{\lambda }+{2\mu n}_{tp2}^{2}+{D}_{1}{\delta }_{fp2}+{D}_{2}{\delta }_{ap2}\right){k}_{tp2}^{2}\right]{A}_{tp2}^{s}\\ {\sigma }_{zz}^{tp3}=\left[{D}_{3}{\delta }_{Tp3}-\left(\overline{\lambda }+{2\mu n}_{tp3}^{2}+{D}_{1}{\delta }_{fp3}+{D}_{2}{\delta }_{ap3}\right){k}_{tp3}^{2}\right]{A}_{tp3}^{s}\\ {\sigma }_{zz}^{tp4}=\left[{D}_{3}{\delta }_{Tp4}-\left(\overline{\lambda }+{2\mu n}_{tp4}^{2}+{D}_{1}{\delta }_{fp4}+{D}_{2}{\delta }_{ap4}\right){k}_{tp4}^{2}\right]{A}_{tp4}^{s}\\ {\sigma }_{zz}^{ts}={2\mu l}_{ts}{n}_{ts}{k}_{ts}^{2}{{\varvec{B}}}_{ts}^{s} \end{array}\right\}$$
(F.1)
$$\left.\begin{array}{c}{\sigma }_{xz}^{tp1}=2\mu {l}_{tp1}{n}_{tp1}{k}_{tp1}^{2}{A}_{tp1}^{s},{\sigma }_{xz}^{tp2}=2\mu {l}_{tp2}{n}_{tp2}{k}_{tp2}^{2}{A}_{tp2}^{s},{\sigma }_{xz}^{tp3}=2\mu {l}_{tp3}{n}_{tp3}{k}_{tp3}^{2}{A}_{tp3}^{s}\\ {\sigma }_{xz}^{tp4}=2\mu {l}_{tp4}{n}_{tp4}{k}_{tp4}^{2}{A}_{tp4}^{s},{\sigma }_{xz}^{ts}=\mu \left({n}_{ts}^{2}-{l}_{ts}^{2}\right){k}_{ts}^{2}{{\varvec{B}}}_{ts}^{s} \end{array}\right\}$$
(F.2)
$$\left.\begin{array}{l}{-p}_{f}^{tp1}=\left[{B}_{4}{\delta }_{Tp1}-\left({B}_{1}+{B}_{2}{\delta }_{fp1}+{B}_{3}{\delta }_{ap1}\right){k}_{tp1}^{2}\right]{A}_{tp1}^{s}\\ {-p}_{f}^{tp2}=\left[{B}_{4}{\delta }_{Tp2}-\left({B}_{1}+{B}_{2}{\delta }_{fp2}+{B}_{3}{\delta }_{ap2}\right){k}_{tp2}^{2}\right]{A}_{tp2}^{s}\\ {-p}_{f}^{tp3}=\left[{B}_{4}{\delta }_{Tp3}-\left({B}_{1}+{B}_{2}{\delta }_{fp3}+{B}_{3}{\delta }_{ap3}\right){k}_{tp3}^{2}\right]{A}_{tp3}^{s}\\ {-p}_{f}^{tp4}=\left[{B}_{4}{\delta }_{Tp4}-\left({B}_{1}+{B}_{2}{\delta }_{fp4}+{B}_{3}{\delta }_{ap4}\right){k}_{tp4}^{2}\right]{A}_{tp4}^{s}\\ {-p}_{f}^{ts}=0 \end{array}\right\}$$
(F.3)
$$\left.\begin{array}{l}{-p}_{a}^{tp1}=\left[{B}_{8}{\delta }_{Tp1}-\left({B}_{5}+{B}_{6}{\delta }_{fp1}+{B}_{7}{\delta }_{ap1}\right){k}_{tp1}^{2}\right]{A}_{tp1}^{s}\\ {-p}_{a}^{tp2}=\left[{B}_{8}{\delta }_{Tp2}-\left({B}_{5}+{B}_{6}{\delta }_{fp2}+{B}_{7}{\delta }_{ap2}\right){k}_{tp2}^{2}\right]{A}_{tp2}^{s}\\ {-p}_{a}^{tp3}=\left[{B}_{8}{\delta }_{Tp3}-\left({B}_{5}+{B}_{6}{\delta }_{fp3}+{B}_{7}{\delta }_{ap3}\right){k}_{tp3}^{2}\right]{A}_{tp3}^{s}\\ {-p}_{a}^{tp4}=\left[{B}_{8}{\delta }_{Tp4}-\left({B}_{5}+{B}_{6}{\delta }_{fp4}+{B}_{7}{\delta }_{ap4}\right){k}_{tp4}^{2}\right]{A}_{tp4}^{s}\\ {-p}_{a}^{ts}=0 \end{array}\right\}$$
(F.4)
$$\left.\begin{array}{l}{\sigma }_{zz}^{ip1}=-\left[\left(3\widehat{\lambda }+2\widehat{\mu }\right){\widehat{\beta }}_{T}{\widehat{\delta }}_{Tp1}+\left(\widehat{\lambda }+2\widehat{\mu }{n}_{ip2}^{2}\right){k}_{ip1}^{2}\right]{A}_{ip1}^{s} \\ {\sigma }_{zz}^{rp1}=-\left[\left(3\widehat{\lambda }+2\widehat{\mu }\right){\widehat{\beta }}_{T}{\widehat{\delta }}_{Tp1}+\left(\widehat{\lambda }+2\widehat{\mu }{n}_{rp1}^{2}\right){k}_{rp1}^{2}\right]{A}_{rp1}^{s}\\ {\sigma }_{zz}^{rp2}=-\left[\left(3\widehat{\lambda }+2\widehat{\mu }\right){\widehat{\beta }}_{T}{\widehat{\delta }}_{Tp2}+\left(\widehat{\lambda }+2\widehat{\mu }{n}_{rp1}^{2}\right){k}_{rp2}^{2}\right]{A}_{rp2}^{s}\\ {\sigma }_{zz}^{rs}=-{2\widehat{\mu }k}_{rs}^{2}{l}_{rs}{n}_{rs} \end{array}\right\}$$
(F.5)
$$\left.\begin{array}{c}{\sigma }_{xz}^{ip1}=2\widehat{\mu }{l}_{ip1}{n}_{ip1}{k}_{ip1}^{2}{A}_{ip1}^{s}, {\sigma }_{xz}^{rp1}=-2\widehat{\mu }{l}_{rp1}{n}_{rp1}{k}_{rp1}^{2}{A}_{rp1}^{s}\\ {\sigma }_{xz}^{rp2}=-2\widehat{\mu }{l}_{rp2}{n}_{rp2}{k}_{rp2}^{2}{A}_{rp2}^{s}, {\sigma }_{xz}^{rs}=\widehat{\mu }\left({n}_{rs}^{2}-{l}_{rs}^{2}\right){k}_{rs}^{2}{{\varvec{B}}}_{rs}^{s}\end{array}\right\}$$
(F.6)
$$\left.\begin{array}{l}{\dot{u}}_{z}^{stp1}=-{n}_{tp1}{v}_{tp1}{k}_{tp1}^{2}{A}_{tp1}^{s},{\dot{u}}_{z}^{stp2}=-{n}_{tp2}{v}_{tp2}{k}_{tp2}^{2}{A}_{tp2}^{s},{\dot{u}}_{z}^{stp3}=-{n}_{tp3}{v}_{tp3}{k}_{tp3}^{2}{A}_{tp3}^{s}\\ {\dot{u}}_{z}^{stp4}=-{n}_{tp4}{v}_{tp4}{k}_{tp4}^{2}{A}_{tp4}^{s},{\dot{u}}_{z}^{sts}={{l}_{ts}{v}_{ts}k}_{ts}^{2}{{\varvec{B}}}_{ts}^{s} \end{array}\right\}$$
(F.7)
$$\left.\begin{array}{l}{\dot{u}}_{x}^{stp1}={l}_{tp1}{v}_{tp1}{k}_{tp1}^{2}{A}_{tp1}^{s},{\dot{u}}_{x}^{stp2}={l}_{tp2}{v}_{tp2}{k}_{tp2}^{2}{A}_{tp2}^{s},{\dot{u}}_{x}^{stp3}={l}_{tp3}{v}_{tp3}{k}_{tp3}^{2}{A}_{tp3}^{s}\\ {\dot{u}}_{x}^{stp4}={l}_{tp4}{v}_{tp4}{k}_{tp4}^{2}{A}_{tp4}^{s},{\dot{u}}_{x}^{sts}={n}_{ts}{v}_{ts}{k}_{ts}^{2}{{\varvec{B}}}_{ts}^{s} \end{array}\right\}$$
(F.8)
$$\left.\begin{array}{l}{\dot{u}}_{z}^{ftp1}=-{n}_{tp1}{v}_{tp1}{k}_{tp1}^{2}{\delta }_{fp1}{A}_{tp1}^{s}, {\dot{u}}_{z}^{ftp2}=-{n}_{tp2}{v}_{tp2}{k}_{tp2}^{2}{\delta }_{fp2}{A}_{tp2}^{s}, {\dot{u}}_{z}^{ftp3}=-{n}_{tp3}{v}_{tp3}{k}_{tp3}^{2}{\delta }_{fp3}{A}_{tp3}^{s}\\ {\dot{u}}_{z}^{ftp4}=-{n}_{tp4}{v}_{tp4}{k}_{tp4}^{2}{{\delta }_{fp4}A}_{tp4}^{s}, {\dot{u}}_{z}^{fts}={v}_{ts}{k}_{ts}^{2}{l}_{ts}{\delta }_{fs}{{\varvec{B}}}_{ts}^{s} \end{array}\right\}$$
(F.9)
$$\left.\begin{array}{l}{\dot{u}}_{z}^{atp1}=-{n}_{tp1}{v}_{tp1}{k}_{tp1}^{2}{\delta }_{ap1}{A}_{tp1}^{s},{\dot{u}}_{z}^{atp2}=-{n}_{tp2}{v}_{tp2}{k}_{tp2}^{2}{\delta }_{ap2}{A}_{tp2}^{s},{\dot{u}}_{z}^{atp3}=-{n}_{tp3}{v}_{tp3}{k}_{tp3}^{2}{\delta }_{ap3}{A}_{tp3}^{s}\\ {\dot{u}}_{z}^{atp4}=-{n}_{tp4}{v}_{tp4}{k}_{tp4}^{2}{{\delta }_{ap4}A}_{tp4}^{s}, {\dot{u}}_{z}^{ats}={v}_{ts}{k}_{ts}^{2}{l}_{ts}{\delta }_{as}{{\varvec{B}}}_{ts}^{s} \end{array}\right\}$$
(F.10)
$$\left.\begin{array}{c}{\dot{u}}_{z}^{sip1}=-{n}_{ip1}{v}_{ip1}{k}_{ip1}^{2}{A}_{ip1}^{s},{\dot{u}}_{z}^{srp1}={n}_{rp1}{v}_{rp1}{k}_{rp1}^{2}{A}_{rp1}^{s},{\dot{u}}_{z}^{srp2}={n}_{rp2}{v}_{rp2}{k}_{rp2}^{2}{A}_{rp2}^{s},{\dot{u}}_{z}^{srs}={l}_{rs}{v}_{rs}{k}_{rs}^{2}{{\varvec{B}}}_{rs}^{s}\\ {\dot{u}}_{x}^{sip1}={l}_{ip1}{v}_{ip1}{k}_{ip1}^{2}{A}_{ip1}^{s},{\dot{u}}_{x}^{srp1}={l}_{rp1}{v}_{rp1}{k}_{rp1}^{2}{A}_{rp1}^{s},{\dot{u}}_{x}^{srp2}={l}_{rp2}{v}_{rp2}{k}_{rp2}^{2}{A}_{rp2}^{s},{\dot{u}}_{x}^{srs}=-{n}_{rs}{v}_{rs}{k}_{rs}^{2}{{\varvec{B}}}_{rs}^{s}\end{array}\right\}$$
(F.11)
