## 3 Analytical Model
### Mechanical Analysis of Lining  
Based on the mechanical model of the lining (Figure 4), the equilibrium equation of the lining is given as follows:  
\[\frac{d\sigma_{r}^{\mathrm{L}}(\tau)}{dr}+\frac{\sigma_{r}^{\mathrm{L}}(\tau)- \sigma_{\theta}^{\mathrm{L}}(\tau)}{r}=0 \tag{19}\]  
where \(\sigma_{r}^{\mathrm{L}}(t)\) and \(\sigma_{\theta}^{\mathrm{L}}(t)\) are the radial and tangential stresses of the lining, respectively.  
The constitutive equations of the lining are expressed as follows:  
\[\left\{\begin{array}{l}\sigma_{r}^{\mathrm{L}}(\tau)=\frac{E^{\mathrm{L}}(r )(1-v^{\mathrm{L}})}{(1+v^{\mathrm{L}})(1-2v^{\mathrm{L}})}\varepsilon_{r}^{ \mathrm{L}}(\tau)+\frac{E^{\mathrm{L}}(r)v^{\mathrm{L}}}{(1+v^{\mathrm{L}})(1- 2v^{\mathrm{L}})}\varepsilon_{\theta}^{\mathrm{L}}(\tau)\\ \sigma_{\theta}^{\mathrm{L}}(\tau)=\frac{E^{\mathrm{L}}(r)(1-v^{\mathrm{L}})}{ (1+v^{\mathrm{L}})(1-2v^{\mathrm{L}})}\varepsilon_{\theta}^{\mathrm{L}}(\tau )+\frac{E^{\mathrm{L}}(r)v^{\mathrm{L}}}{(1+v^{\mathrm{L}})(1-2v^{\mathrm{L}} )}\varepsilon_{r}^{\mathrm{L}}(\tau)\end{array}\right. \tag{20}\]  
where \(E^{\mathrm{L}}(r)\) and \(v^{\mathrm{L}}\) are the elastic modulus and the Poisson's ratio of the lining, respectively; and \(\varepsilon_{r}^{\mathrm{L}}(t)\) and \(\varepsilon_{\theta}^{\mathrm{L}}(t)\) are the radial and tangential strains of the lining, respectively.  
The geometric equations of the lining are given as follows:  
\[\left\{\begin{array}{l}\varepsilon_{r}^{\mathrm{L}}(\tau)=\frac{du_{r}^{ \mathrm{L}}(\tau)}{dr}\\ \varepsilon_{\theta}^{\mathrm{L}}(\tau)=\frac{dv_{r}^{\mathrm{L}}(\tau)}{r} \end{array}\right. \tag{21}\]  
where \(u_{r}^{\mathrm{L}}(\tau)\) is the radial deformation of the lining.  
As mentioned above, the elastic modulus of the lining is a function of the radial coordinate. In this study, this function is assumed to be a power function defined as follows:  
\[E^{\mathrm{L}}(r)=E_{0}^{\mathrm{L}}r^{\delta} \tag{22}\]  
where \(E^{\mathrm{L}}_{0}\) is the initial value of the elastic modulus of the lining; and \(\delta\) is the radially inhomogeneous coefficient of the lining. When \(\delta\) is equal to zero, the inhomogeneous degree of the lining is equal to zero, indicating that the lining is a homogeneous material.  
Substituting Equations (20)-(22) into Equation (19), the governing equation in terms of radial deformation of the lining can be obtained as follows:  
\[\frac{d^{2}u_{r}^{\mathrm{L}}(\tau)}{dr^{2}}+\frac{\delta+1}{r}\frac{du_{r}^{ \mathrm{L}}(\tau)}{dr}+\frac{(\delta+1)v^{\mathrm{L}}-1}{(1-v^{\mathrm{L}})r^{ 2}}u_{r}^{\mathrm{L}}(\tau)=0 \tag{23}\]  
Figure 4: Plan sketch of the mechanical model of the lining.  
The boundary conditions of the lining are set as follows:  
\[\left\{\begin{array}{l}\left.\left[\sigma_{r}^{\mathrm{L}}(\tau)\right]\right|_{ r=R_{1}}=q(\tau)\\ \left.\left[\sigma_{r}^{\mathrm{L}}(\tau)\right]\right|_{r=R_{2}}=Q(\tau)\end{array}\right. \tag{24}\]  
Based on the boundary conditions, the solution of the governing equation Equation (23) can be obtained as follows:  
\[u_{r}^{\mathrm{L}}(\tau)=R_{2}[A_{0}E(r)q(\tau)+A_{0}F(r)Q(\tau)] \tag{25}\]  
in which  
\[\left\{\begin{array}{l}A_{0}=\frac{1}{R_{2}}\frac{1}{W(R_{1})M(R_{2})-W(R_{2 })M(R_{1})}\\ E(r)=M(R_{2})r^{a_{1}}-W(R_{2})r^{a_{2}}\\ F(r)=W(R_{1})r^{a_{2}}-M(R_{1})r^{a_{1}}\\ W(r)=r^{\beta+a_{1}-1}\left[\frac{a_{1}E_{1}^{b}(1-r^{\mathrm{L}})}{(1+r^{ \mathrm{L}})(1-2r^{\mathrm{L}})}+\frac{E_{0}^{\mathrm{L}}v^{\mathrm{L}}}{(1+ r^{\mathrm{L}})(1-2r^{\mathrm{L}})}\right]\\ M(r)=r^{\beta+a_{2}-1}\left[\frac{a_{2}E_{1}^{b}(1-r^{\mathrm{L}})}{(1+r^{ \mathrm{L}})(1-2r^{\mathrm{L}})}+\frac{E_{0}^{\mathrm{L}}v^{\mathrm{L}}}{(1+ r^{\mathrm{L}})(1-2r^{\mathrm{L}})}\right]\\ a_{1}=-\frac{\delta-\sqrt{(\delta)^{2}-4\delta\theta+4}}{2}\\ a_{2}=-\frac{\delta+\sqrt{(\delta)^{2}-4\delta\theta+4}}{2}\\ \vartheta=\frac{v^{\mathrm{L}}}{1-v^{\mathrm{L}}}.\end{array}\right. \tag{26}\]