## 2 Factors Controlling Friction Forces
### Earth Pressure around the Box  
The over-cutting space causes the soil to loosen, creating a new stress state. The Terzaghi loose soil pressure theory considers the restraining effect of the soil column around the tunnel on the overlying soil mass, known as the soil arching effect. This concept has been adopted by circular jacking specifications in various countries [14; 15]. As shown in Figure 3, the Terzaghi model assumes that part of the soil pressure in the shear zone is balanced by shear resistance. The earth pressure on the box can be calculated according to Equation (1). \[ \begin{split}\sigma_{\text{z}}=\frac{1-\text{e}^{-2K\tan\varphi \frac{Z}{B_{1}}}}{2K\tan\varphi\frac{Z}{B_{1}}}\left(\gamma-\frac{2c}{B_{1}} \right)Z+q_{0}\text{e}^{-2K\tan\varphi\frac{Z}{B_{1}}}\\ B_{1}=B+2H\tan(45^{\circ}-\varphi/2)\end{split}  \tag{1}\]  
where \(\sigma_{z}\)--earth pressure (kPa);  
\(c\)--the cohesion (kPa);  
\(\varphi\)--the internal friction angle (\({}^{\circ}\));  
\(B_{1}\)--the the shear zone width (m);  
\(B\)--the box width (m);  
\(H\)--the box height (m);  
\(q_{0}\)--the additional load on the ground (kPa);  
\(\gamma\)--the unit weight of the soil (kN/m\({}^{3}\));  
\(Z\)--the box buried depth (m);  
\(K\)--the earth pressure coefficient, which numerous experiments and simulation studies suggest setting it to 1. Equation (1) is applicable for calculating the soil pressure under conditions without lubricants. This paper provides a correction considering the effect of the lubricant. When the gate moves downward relative to the stationary soil, shear stress is generated upward at the shear interface, known as positive soil arching. When the gate moves upward relative to the stationary soil, shear stress is generated downward at the shear interface, known as  
Figure 2: The box-soil contact status evolutions. negative soil arching [16]. As shown in Figure 4, during the repeated injection and loss of lubricant, the box continuously floats up and sinks. Therefore, the soil pressure at the top of the box is controlled between positive and negative soil arching. Meanwhile, [PERSON] [15] introduced the non-associated plastic flow rule into the soil shear strain problem, as shown in Equation (2):  
\[\tau=\sigma\frac{\cos\psi\sin\varphi}{1-\sin\psi\sin\varphi} \tag{2}\]  
Using on-site measurements, numerical simulations, and PIV experiments, research on the parameters of the dilation angle (\(\psi\)) has demonstrated that it can be set to \(0^{\circ}\)[17; 18]. Therefore, Equation (2) can be simplified to \(\tau=\sigma\mathrm{sin}\varphi\). Figure 4: Effect of lubricant on soil arching direction. Figure 3: Terzaghi loose soil pressure calculation model. (1) Earth pressure in the positive soil arching state. As shown in Figure 4a, as the lubricant gradually dissipates, the soil within the shear zone slides downward relative to the soil on both sides. Shear stress acts upward to inhibit its sliding tendency, presenting a positive soil arching state, and the earth pressure reaches a minimum. Considering the force transmission via particle compression between layered soils, the bottom earth pressure of the upper layer is defined as the additional load on the top of the lower layer, as shown in Equation (3):  
\[ \begin{split}\sigma_{znt}=\frac{1-e^{-2\sin\frac{\varphi_{2n}}{ \Phi_{1}}}}{2\sin\varphi\frac{\varphi_{2n}}{\Phi_{1}}}\Big{(}\gamma_{n}-\frac {2c}{\bar{B}_{1}}\Big{)}Z_{n}+\sigma_{znt-1}e^{-2\sin\varphi\frac{\varphi_{2n}} {\bar{B}_{1}}}\\ B_{1}=B+2H\tan(45^{\circ}-\varphi/2)\end{split}  \tag{3}\]  
where \(\sigma_{znt}\)--the earth pressure in the \(n\)th layer (kPa);  
\(Z_{n}\)--the soil thickness in the \(n\)th layer (m). (2) Earth pressure in the negative soil arching state. As shown in Figure 4b, the lubricant volume in the over-cutting space gradually increases during the lubricant injection. The formed mud cake seals the filtration channel, causing the lubricant pressure to act directly on the box and the soil around the tunnel.