# Temperature Effect on Stability of Tunnel Face Under Unsaturated Seepage Condition
## 3 Problem Formulation  
### Three-Dimensional Logarithmic Spiral Failure Mechanism  
The study presented herein is depicted in Figure 1. The research bases on a tunnel devoid of face support stresses, having a diameter of \(D\) and an overburden depth of \(C\). The surrounding soil experiences unsaturated flow. A horizontal plane is set at a vertical distance \(z_{0}\) above the tunnel's base. This research utilizes the kinematic limit analysis approach, prioritizing the development of a viable collapse mechanism. The assumption is made that the soil adheres to the associative flow rule, indicating that the direction of the velocity vector should coincide  
Figure 1: Schematic of the failure mechanism. with the effective internal friction angle at the failure surface. Consequently, \(v_{\rm n}\) can be formulated as  
\[v_{\rm n}=v_{\rm t}{\rm tan}\theta^{\prime} \tag{14}\]  
In this context, \(v_{\rm n}\) signifies the normal velocity component on the sliding surface, and \(v_{\rm t}\) refers to the tangential velocity component. This formula demonstrates that the mechanism's outline takes the shape of a logarithmic spiral during rigid rotation, with the sliding surface touching a cone that has a vertex angle of \(2\theta^{\prime}\). Employing the 3D logarithmic spiral failure mechanism to describe the tunnel face collapse meets the aforementioned conditions. The mechanism rotates with an angular velocity \(\omega\) about the central point O. As a result, a polar coordinate framework \((\theta,\rho)\) is set up to determine the velocity at any given point within the mechanism:  
\[v=\omega\rho \tag{15}\]  
The mechanism's trajectory is governed by two logarithmic spiral paths, specifically the AE and BE curves, which are formulated as follows:  
\[\left\{ \begin{array}{l}r_{\rm AE}=r_{\rm A}{\rm e}^{-(\theta-\theta_{\rm A} ){\rm tan}\varphi^{\prime}}\\ r_{\rm BE}=r_{\rm BE}({\rm e}^{(\theta-\theta_{\rm B}){\rm tan}\varphi^{ \prime}}\end{array} \right. \tag{16}\]  
where \(r_{\rm A}\) and \(r_{\rm B}\) can be represented by \(\theta_{\rm A}\) and \(\theta_{\rm B}\):  
\[\left\{ \begin{array}{l}r_{\rm A}=\frac{{\rm sin}\theta_{\rm B}}{{\rm sin}( \theta_{\rm B}-\theta_{\rm A})}D\\ r_{\rm B}=\frac{{\rm sin}\theta_{\rm A}}{{\rm sin}(\theta_{\rm B}-\theta_{\rm A })}D\end{array} \right. \tag{17}\]  
Substitute Equation (17) into Equation (16):  
\[\left\{ \begin{array}{l}r_{\rm AE}=\frac{D{\rm sin}\theta_{\rm B}}{{\rm sin}( \theta_{\rm B}-\theta_{\rm A})}{\rm e}^{-(\theta-\theta_{\rm A}){\rm tan} \varphi^{\prime}}\\ r_{\rm BE}=\frac{D{\rm sin}\theta_{\rm A}}{{\rm sin}(\theta_{\rm B}-\theta_{ \rm A})}{\rm e}^{(\theta-\theta_{\rm B}){\rm tan}\varphi^{\prime}}\end{array} \right. \tag{18}\]  
The mechanism is composed of several circles centered at points along the rotation axis of the two logarithmic spiral curves. The trajectory of the central point is formulated as  
\[r_{\rm m}=\frac{r+r^{\prime}}{2} \tag{19}\]  
The radius of these circles can be calculated using the following approach:  
\[R=\frac{r-r^{\prime}}{2} \tag{20}\]  
The meeting point of the dual logarithmic spiral curves is at point E, identifiable with polar coordinates:  
\[\left\{ \begin{array}{l}\theta_{\rm E}=\frac{\theta_{\rm A}+\theta_{\rm B}} {2}+\frac{{\rm ln}({\rm sin}\theta_{\rm B}/{\rm sin}\theta_{\rm A})}{2{\rm tan }\varphi^{\prime}}\\ r_{\rm E}=\frac{\sqrt{{\rm sin}\theta_{\rm A}{\rm sin}\theta_{\rm B}}}{{\rm sin }(\theta_{\rm B}-\theta_{\rm A})}{\rm e}^{\frac{1}{2}(\theta_{\rm A}-\theta_{ \rm B}){\rm tan}\varphi^{\prime}}D\end{array} \right. \tag{21}\]  
In the case where the failure mechanism outcrops, curves AE and BE intersect the ground at points C and D, respectively. The mechanism is controlled by curves \(AC\), \(BD\), and ground CD. The positions of points C and D can be represented as  
\[\left\{