## 4 Deformation Mechanism
### Deformation Mechanism
where \(k\) is used to describe the extent of the volume change of the soil, and \(q_{s}\) denotes the flux through plane \(\Pi_{\beta}\). At the tunnel face, \(q_{s}\) is equal to \(q_{\textit{sf}}\). \(q_{s}\) equal to \(Q_{s}\) at the ground surface. \[ \begin{array}{ll}q_{s}&=\int_{0}^{2\pi}\int_{0}^{R_{\max}(\theta,\beta)}v_{ \beta}(r,\,\theta,\beta)r\mathrm{d}r\mathrm{d}\theta\\ &=\int_{0}^{2\pi}\int_{0}^{R_{\max}(\theta,\beta)}v_{m}(\beta)\Big{(}1-\frac{ r^{2}}{R_{\max}^{2}(\theta,\,\beta)}\Big{)}r\mathrm{d}r\mathrm{d}\theta\\ &=v_{m}(\beta)\int_{0}^{2\pi}\int_{0}^{R_{\max}(\theta,\beta)}\Big{(}1-\frac{ r^{2}}{R_{\max}^{2}(\theta,\,\beta)}\Big{)}r\mathrm{d}r\mathrm{d}\theta\\ &=v_{m}(\beta)\frac{\tau D^{2}\Omega_{1}^{2}}{8\Omega_{3}^{2}}\end{array}  \tag{15}\]  
So, combining Equations (14) and (15), the maximum orthoradial displacement on any plane can be given by Equation (16). \[v_{m}(\beta)=\frac{S_{\mathrm{mf}}(k\beta+1)\Omega_{3}^{2}}{\Omega_{1}^{2}} \tag{16}\]  
So, \(v_{\beta}\) can be derived as follows. \[v_{\beta}(r,\,\theta,\,\beta)=\frac{S_{\mathrm{mf}}(k\beta+1)\Omega_{3}^{2}}{ \Omega_{1}^{2}}\bigg{(}1-\frac{r^{2}}{R_{\max}^{2}(\theta,\,\,\beta)}\bigg{)} \tag{17}\]  
In solid mechanics, there are two expressions for the volumetric strain (\(\Theta\)): the ratio of the volume change (\(\Delta V\)) to the original volume (\(V\)), and the summation of the normal strains in three mutually perpendicular directions, as shown in Equation (18). \[\Theta=\frac{\Delta V}{V}=\varepsilon_{r}+\varepsilon_{\theta}+\varepsilon_{\beta} \tag{18}\]  
Moreover, the volume change (\(\Delta V\)) of the \(\Pi_{\beta}\) plane with \(\mathrm{d}\beta\) can be calculated by taking the derivative of \(q_{s}\) as shown in Equation (19). \[\Delta V=\frac{\partial q_{s}}{\partial\beta}\cdot\mathrm{d}\beta=q_{s}/k \cdot\mathrm{d}\beta \tag{19}\]  
The original volume of the \(\Pi_{\beta}\) plane with \(\mathrm{d}\beta\) can be given as follows:  
\[V=\mathrm{d}\beta\cdot R_{f}\cdot D_{s}(\beta)^{2}\pi/4 \tag{20}\]  
where \(D_{S}(\beta)\) is the diameter of the \(\Pi_{\beta}\) plane, and can be expressed as follows. \[D_{s}(\beta)=R_{\max}(\theta=0,\,\,\beta)+R_{\max}(\theta=\frac{\pi}{2},\,\, \beta)=\frac{\Omega_{1}D}{\Omega_{3}} \tag{21}\]  
So, we have:  
\[\Theta=\frac{S_{\mathrm{mf}}\cdot k\Omega_{3}^{2}}{2R_{\mathrm{f}}\Omega_{1}^ {2}} \tag{22}\]  
The second expression of the volumetric volume shown in Equation (18) can be calculated by the strain introduced in Section 4.2. The volumetric volume can be also given as follows. \[\frac{\partial v_{r}}{\partial r}+\frac{1}{r}v_{r}+\frac{1}{R_{\beta}-r\cos \theta}(-v_{r}\cos\theta+\frac{\partial v_{\beta}}{\partial\beta})=\Theta \tag{23}\]  
After simplification of Equation (23), we have:  
\[\frac{\partial v_{r}}{\partial\tau}+\frac{(\overline{R}_{\beta}-2\overline{r} \cos\theta)}{\overline{r}(\overline{R}_{\beta}-\overline{r}\cos\theta)}v_{r }=-\frac{1}{(\overline{R}_{\beta}-\overline{r}\cos\theta)}\frac{\partial v_{ \beta}}{\partial\beta}+\frac{S_{\mathrm{mf}}\cdot k\Omega_{3}^{2}}{2\overline{ R}_{\beta}\Omega_{1}^{2}} \tag{24}\]The derivative of orthoradial displacement \(v_{\beta}\) with respect to \(\beta\) can be given as follows:  
\[ \begin{array}{ll}\frac{\partial v_{\beta}}{\partial\beta}&=S_{\rm mf}\frac{ \Omega_{2}^{2}}{\Omega_{1}^{2}}\left[k\bigg{(}1-\frac{4\Omega_{2}^{2}\eta^{2}} {\Omega_{1}^{2}\Omega_{2}^{2}}\bigg{)}+(k\beta+1)\bigg{(}-2\Omega_{4}-\frac{4 \Omega_{2}^{2}\Omega_{5}\eta^{2}}{\Omega_{1}^{2}\Omega_{2}^{2}}\bigg{)}\right] \\ &=S_{\rm mf}\frac{\Omega_{2}^{2}}{\Omega_{1}^{2}}\bigg{[}-4\frac{\Omega_{2}^{2} }{\Omega_{1}^{2}\Omega_{2}^{2}}\Big{(}k+\frac{(k\beta+1)\Omega_{5}}{\Omega_{1 }\Omega_{2}}\Big{)}\vec{r}^{2}+k-2(k\beta+1)\Omega_{4}\bigg{]}\end{array}  \tag{25}\]  
\[\Omega_{5}(\theta,\beta)=-4\Omega_{2}\Omega_{4}+3\Omega_{1}\Bigg{(}\cos\theta+ \frac{2\overline{L}\sin^{2}\theta}{\sqrt{1-2\overline{L}^{2}+2\overline{L}^{2 }\cos 2\theta}}\Bigg{)} \tag{26}\]  
Equation (24) is an ordinary first-order linear differential equation.