## 2 Mechanical Model Establishment and Equations Derivation
### Calculation of Soil Displacement at Shield Tunnel [PERSON] [14] proposed the virtual image technique in 1987 and derived the equation of the distribution law of the surrounding soil displacement field caused by stratum loss at any point in the elastic half-space. As shown in Figure 3, the basic analysis steps are as follows:  
1. Ignoring the surface of the semi-infinite space, this hole is called a positive source, and the problem will be transformed into the space under the infinite space. The solution of the displacement field of the soil layer in the gap will generate the normal stress \(\sigma_{0}\) and the shear stress \(\tau_{0}\) on the surface of the infinite space. 2. A fictitious volume expansion of the same size is set at the mirror image position of the void in the infinite space, and the opposite is called a negative source hole. The symmetrical negative source will generate positive stress \(\sigma_{0}\) and shear stress \(\tau_{0}\) on the original infinite surface. Under the action of the positive source hole and the negative source hole, the normal stress on the original surface that does not exist can be offset. 3. The positive and negative source holes above are superposed, and the normal stress counteracts each other, while the shear stress is converted to \(2\tau_{0}\). When the shear stress \(2\tau_{0}\) is inversely applied to the surface of the original semi-infinite body, and the above three steps are superimposed, the solution of the displacement field under the layer loss on the surface of the semi-infinite body can be obtained. Figure 3: Analysis flow chart of the virtual image technique. Based on this method, this study calculates the soil displacement because of the deformation of the enclosure structure. It adopts a certain simplification of the displacement mechanism of the soil loss. The hypothesizes are as follows:  
1. No consideration of pore water pressure and soil consolidation;
2. The soil is incompressible, and the soil displacement field of the is only caused by the soil loss;
3. Each section of the envelope structure deforms in the same size. The calculation model of the virtual image technique is shown in Figure 4. The displacement at point \(p\) (\(x_{1},z_{1}\)) on the tunnel axis along the \(X\)-axis produced by the sink of the radius \(a\) at point \(F\) (\(B/2\), \(\eta\)) is:  
\[S_{\text{x1}}=-\frac{a^{2}}{2}\frac{x_{1}-B/2}{{r_{1}}^{2}} \tag{4}\]  
where \(r_{1}=\sqrt{\left(B/2-x_{1}\right)^{2}+\left(\eta-z_{1}\right)^{2}}\), indicates the distance between point \(F\) and point \(p\). Since Equation (4) is the displacement expression of infinite space, but the construction site is a semi-infinite space; it is necessary to transform the problem of solving the infinite space into the problem of solving a semi-infinite space. Therefore, the point \(F\) (\(B/2\), \(\eta\)) is mirrored as point \(F\) (\(B/2\), \(-\eta\)), where an equal volume expansion occurs. The displacement at point \(p\) (\(x_{1},z_{1}\)) on the tunnel axis along the \(X\)-axis produced by the source of the radius \(a\) at point \(F\) (\(B/2\), \(-\eta\)) is:  
\[S_{\text{x2}}=\frac{a2}{2}\frac{x_{1}-B/2}{r_{2}2} \tag{5}\]  
where \(r_{2}=\sqrt{\left(B/2-x_{1}\right)^{2}+\left(\eta+z_{1}\right)^{2}}\), which indicates the distance between point \(F\) and point \(p\). During the establishment of Equation (4), the shear strain \(\gamma\) will be generated on the ground:  
\[\gamma\text{=}-4a^{2}\frac{\eta(x_{1}-B/2)}{\left[\left(x_{1}-B/2\right)^{2}+ \eta^{2}\right]^{2}} \tag{6}\]  
Then, the additional shear stress generated on the surface is:  
\[\tau=G\gamma\text{=}-4 Ga^{2}\frac{\eta(x_{1}-B/2)}{\left[\left(x_{1}-B/2\right) ^{2}+\eta^{2}\right]^{2}} \tag{7}\]  
where \(G\) is the shear modulus of the soil. Figure 4: Calculation model of the virtual image technique.