## 3 Analytical Model
### Deformation Compatibility on Rock-Lining Interface
Then, \(Q_{4}(\tau)\) can be deduced by the Laplace transform inversion about parameter \(s\) of the \(f(s)\). Subsequently, the corresponding solutions of \(Q_{4}(\tau)\) are derived as follows:  
(a) When the inner surface pressure acting on the lining along the radial direction is constant, the expression of \(q(\tau)\) can be given by:  
\[q(\tau)=W \tag{36}\]  
where \(W\) is the water pressure. Using Laplace transform, the expression of the \(q(s)\) can be given by:  
\[q(s)=\frac{W}{s} \tag{37}\]  
Substituting Equation (37) into Equation (34) (see expression \(f(s)\)),\(f(s)\) can be obtained as follows:  
\[f(s)=\frac{2 WA_{0}E(R_{2})(s+1/t_{k})}{a_{11}(s-x_{1})(s-x_{2})} \tag{38}\]  
Using the Laplace transform inversion, the expression of \(Q_{4}(\tau)\) can be derived as follows:  
\[Q(\tau)_{4}=\frac{2A_{0}E(R_{2})W[e^{xx_{1}}(1+t_{k}x_{1})-e^{\tau x_{2}}(1+t_ {k}x_{2})]}{a_{11}t_{k}(x_{1}-x_{2})} \tag{39}\]  
(b) When the inner surface pressure acting on the lining along the radial direction changes, the expression of \(q(\tau)\) can be given as follows:  
\[q(\tau)=F_{0}e^{-\zeta\tau}\sin(\chi\tau+\psi) \tag{40}\]where \(F_{0}\) is the initial amplitude of the aerodynamic pressure; \(\zeta\) is the damping coefficient of the aerodynamic pressure; \(\chi\) is the angular frequency of the aerodynamic pressure; and \(\psi\) is the initial phase angle of the aerodynamic pressure. Using Laplace transform, the expression of \(q(s)\) can be given as follows:  
\[q(s)=F_{0}\frac{\sin(\psi)(\zeta+s)+\chi\cos(\psi)}{\left(\zeta+s\right)^{2}+ \chi^{2}} \tag{41}\]  
Substituting Equation (41) into Equation (34) (see expression \(f(s)\)), \(f(s)\) can be obtained as follows:  
\[f(s)=\frac{2F_{0}A_{0}E(R_{2})s(s+1/t_{\rm k})}{a_{11}(s-x_{1})(s-x_{2})}\frac{ \sin(\psi)(\zeta+s)^{2}+\chi\cos(\psi)}{\left(\zeta+s\right)^{2}+\chi^{2}} \tag{42}\]  
Using the Laplace transform inversion, \(Q_{4}(\tau)\) can be expressed as follows:  
\[ \begin{array}{ll}Q(\tau)_{4}&=\frac{2A_{0}E(R_{2})F_{0}}{a_{11}t_{d}(x_{1}- x_{2})}\Big{\{}\frac{x_{1}e^{x_{1}}[(\chi+\chi t_{d}x_{1})\cos(\psi)+H_{10}\sin( \psi)]}{R_{8}}-\frac{x_{2}e^{x_{2}}[(\chi+\chi t_{d}x_{2})\cos(\psi)+H_{11} \sin(\psi)]}{H_{9}}\Big{\}}\\ -\frac{e^{-\zeta t_{L}}L_{1}\big{[}\cosh(\chi t_{1}t_{1})+\frac{\sin(\chi t_{1 }t_{2})\ln{\frac{1}{\chi}}}{\lambda}\big{]}}{a_{11}t_{d}B_{h}B_{9}}\end{array}  \tag{43}\]  
in which  
\[ \begin{array}{ll}&H_{1}=\zeta^{4}+\zeta^{3}x_{1}+\zeta^{3}x_{2}+\zeta x^{2}x _{1}+\zeta x^{2}x_{2}+\zeta^{2}x_{1}x_{2}+\chi^{2}x_{1}x_{2}\\ &H_{2}=2\zeta^{3}_{x}\zeta^{2}t_{d}+\chi^{4}+\zeta^{5}t_{d}+\zeta\chi t^{4}t_{ d}+\zeta^{4}t_{d}x_{1}+\zeta^{4}t_{d}x_{2}+\chi^{4}t_{d}x_{1}+\chi^{4}t_{d}x_{2}\\ &+\zeta^{3}t_{d}x_{1}x_{2}+2\zeta^{2}\chi^{2}t_{d}x_{1}+2\zeta^{2}\chi^{2}t_{ d}x_{2}+\zeta\chi^{2}t_{d}x_{1}x_{2}\\ &H_{3}=2\zeta\chi^{3}+2\zeta^{3}\chi+\chi^{2}x_{1}+\chi^{3}x_{2}+\zeta^{2} \chi x_{1}+\zeta^{2}\chi x_{2}+\chi^{3}t_{d}x_{1}x_{2}+\zeta^{2}\chi t_{d}x_{1} x_{2}\\ &H_{4}=\chi^{5}t_{d}+\zeta^{4}\chi t_{d}+2\zeta^{2}\chi^{3}t_{d}\\ &H_{5}=\chi^{3}+\zeta^{5}\chi t_{d}x_{1}+\zeta^{2}\chi t_{d}x_{2}+\chi^{3}t_{d} x_{1}+\chi^{3}t_{d}x_{2}+\zeta^{2}\chi+2\zeta\chi t_{d}x_{1}x_{2}-\chi x_{1}x_{2}\\ &H_{6}=\zeta^{3}+\zeta\chi^{2}+\zeta^{2}x_{1}+\zeta^{2}x_{2}+\chi^{2}x_{1}+ \chi^{2}x_{2}+\zeta x_{1}x_{2}+\chi^{2}t_{d}x_{1}x_{2}\\ &H_{7}=\zeta^{4}t_{d}-\chi^{4}t_{d}-2\zeta^{2}\chi^{2}t_{d}-\zeta^{3}t_{d}x_{1} -\zeta^{3}t_{d}x_{2}-\zeta\chi^{2}t_{d}x_{1}-\zeta\chi^{2}t_{d}x_{2}-\zeta^{2 }t_{d}x_{1}x_{2}\\ &H_{8}=\zeta^{2}+2\zeta x_{1}+\chi^{2}+x_{1}^{2}\\ &H_{9}=\zeta^{2}+2\zeta x_{2}+\chi^{2}+x_{2}^{2}\\ &H_{10}=\zeta+x_{1}+t_{d}x_{1}^{2}+\zeta t_{d}x_{1}\\ &H_{11}=\zeta+x_{2}+t_{d}x_{2}^{2}+\zeta t_{d}x_{2}\\ &L_{1}=2A_{0}E(R_{2})F_{0}|H_{5}\cos(\psi)+\left(H_{6}-H_{7}\right)\sin(\psi) \right]\\ &L_{2}=\frac{\left(H_{1}-H_{2}\right)\sin(\psi)+\left(H_{3}-H_{4}\right)\cos( \psi)}{H_{5}\cos(\psi)+\left(H_{6}-H_{7}\right)\sin(\psi)}\end{array}  \tag{44}\]