## 3 Principle of Minimum Total Potential Energy  
The principle of minimum total potential energy is widely utilized in different industries, including chemistry, physics, and engineering. Moreover, many numerical simulation methods are based on it. The principle of minimum total potential energy indicates that an unbalanced system will finally translate to a balanced system with the minimum total potential energy. In order to utilize the minimum total potential energy principle in the proposed problem, the start and end of the deformation process first have to be determined. As shown in Figure 2, the soil is stable in the initial unbalanced state. When tunnelling, the support pressure supplied by slurry cannot substitute the original earth pressure, and face stress relief happens. As a result, the soil begins to move into the excavation chamber. In our model, the support pressure was assumed to be constant during the deformation process. At last, the system was stable when the total potential energy reached the minimum potential energy in the final equilibrium state. So, we determined the final equilibrium by finding the state with the minimum total potential energy. In other words, the parameters in the deformation mechanism can be determined by minimizing the total potential energy of the studied system.  
In our model, the moved soil is regarded as the studied system in our method. The potential energy of the studied system contains three components: elastic strain energy stored in the deformed system (\(U_{s}\)), the potential energy of gravity (\(V_{g}\)), and the potential energy of the support pressure (\(V_{p}\)). \(V_{g}\) and \(V_{p}\) belong to the potential energy of external force associated with the external force (V).  
The elastic strain energy can be calculated by the strain of the soil. In elastic mechanics, the strain energy density (strain energy of unit volume) can be calculated as follows:  
\[U_{s0}=\frac{E}{2(1+\mu)}\bigg{[}\frac{\mu}{1-2\mu}(\varepsilon_{x}+\varepsilon _{y}+\varepsilon_{z})^{2}+(\varepsilon_{x}^{2}+\varepsilon_{y}^{2}+\varepsilon _{z}^{2})+\frac{1}{2}\Big{(}\gamma_{xy}^{2}+\gamma_{xz}^{2}+\gamma_{yz}^{2} \Big{)}\bigg{]} \tag{3}\]  
where \(\varepsilon_{x}\), \(\varepsilon_{y}\), \(\varepsilon_{z}\), \(\varepsilon_{xy}\), \(\varepsilon_{xz}\), and \(\varepsilon_{yx}\), are the strain in the Cartesian coordinate system as shown in Figure 2, \(E\) is the elastic modulus of the soil, and \(\mu\) is Poisson's ratio of the soil. Furthermore, the soil is assumed to behave elastically. The strain energy can be calculated by integrating \(U_{s0}\) over the moved soil:  
\[V_{P}=\int_{N}U_{s0}\mathrm{d}n \tag{4}\]  
where \(N\) is the volume of the moved soil in the deformation mechanism.  
The external force consists of the support pressure and gravity. The potential energy of the external force is generally measured by the work performed by the external force. The increase in the potential energy of the external force is equal to the minus work performed by the external force. So, the potential energy of the support pressure and gravity can be given as follows:  
\[\left\{\begin{array}{l}V_{P}=-\int_{S}\left(P-\gamma_{w}\right)u_{f}\mathrm{ d}s\\ V_{g}=-\int_{N}g^{\prime}u\mathrm{d}n\end{array}\right. \tag{5}\]  
where \(u_{f}\) is the axial displacement on the face, \(w\) is the vertical displacement of the soil, and \(S\) is the area of the face.  
Some parameters for the calculation of potential energy (\(w\), \(u_{f}\), and the strain of soil) are tightly related to the deformation mechanism. Thus, these parameters are introduced in Section 4.2 after the deformation mechanism introduction.