## 2 Convergence-Confinement Method and Displacement Release Coefficient
### Displacement Release Coefficient Analysis  
For complex elastic-plastic problems, the relationship between stress and strain no longer satisfies the simple hoke equation, but depends on the plastic constitutive relationship of the material [15]. In order to more clearly and intuitively study the \"spatial effect\" of the tunnel face, the displacement release coefficient \(\lambda(x)\) is defined to describe the \"spatial effect\" of the face. The displacement release coefficient \(\lambda(x)\) is the ratio of the radial displacement \(u(x)\) of the surrounding rock at a certain point, at a certain distance \(x\) from the face to the radial displacement \(u(\infty)\), at the same position and in the same direction, where the face is far enough away (not affected by the spatial constraint effect of the face):  
\[\lambda(x)\ =\ u(x)/u(\infty) \tag{1}\]  
The radial displacement \(u(\infty)\) of surrounding rock is calculated by empirical Equation:  
\[u(\infty)\ =\ P_{0}R/2G \tag{2}\]  
where \(P_{0}\) is the initial in situ stress, \(R\) is the tunnel excavation radius and \(G\) is the shear modulus. Whether the displacement release coefficient \(\lambda\) can be effectively applied to calculate the longitudinal deformation curve of surrounding rock is the focus of current scholars. At present, the calculation equation of the displacement release coefficient \(\lambda\) in representing the longitudinal deformation curve of surrounding rock is as follows:  
[PERSON] [16] obtained the relationship of the displacement release coefficient when \(x\geq 0\) by using three-dimensional finite element elastic analysis (Equation (3)):  
\[\lambda(x)\ =\ 0.25+0.75\Big{[}1-(0.75/(0.75+x/R))^{2}\Big{]} \tag{3}\]  
[PERSON] [17] used the elastic stress analysis method to put forward the relationship of the displacement release coefficient when \(x=0\), as in the following Equation:  
\[\lambda(x)\ =\ 0.29+0.71\Big{\{}1-\exp\Bigl{[}-1.5(x/R)^{0.7}\Big{]}\Big{\}} \tag{4}\]  
The displacement release coefficients of the two equations at the tunnel face are 0.25 and 0.29, respectively. It is only suitable for section \(x\geq 0\), which is contrary to the actual project. Plastic deformation often occurs in the surrounding rock in the actual underground engineering construction. It is more important to study the displacement release coefficient in the plastic deformation stage of the surrounding rock. The relationship between the displacement release coefficient of surrounding rock under the condition of elastic-plastic deformation is as follows:  
[PERSON] [18], based on the elastic-plastic model and H-B strength criterion, uses FLAC3D for the simulation calculation to construct the V-D (09) displacement release coefficient based on plastic radius \(R\) (Equation (5)):  
\[\lambda(x)\ =\ \left\{ \begin{array}{l}0.34\exp(-0.15R^{*}),x\ =\ 0\\ \lambda_{0}\exp(x^{*}),x\leq 0\\ 1-(1-\lambda_{0})\exp(-0.15x^{*}/R^{*}),x\geq 0\end{array} \right. \tag{5}\]  
where the relative radius \(R^{*}=R/r\), relative position \(x^{*}=x/r\) and \(\lambda_{0}\) is the displacement release coefficient at \(x=0\) from the tunnel face. [PERSON] [19] used the normalization method to analyze the tunnel monitoring data and proposed the empirical displacement release coefficient equation (Equation (6)) where its \(\lambda_{0}=0\). \[\lambda(x)\ =\ \frac{1}{2}\bigg{[}1-\tanh\bigg{(}\frac{1}{3}-\frac{x}{2\text{R }}\bigg{)}\bigg{]} \tag{6}\]  
Here, \(x\) is the axial distance from the excavation surface and R is half of the tunnel excavation diameter. [PERSON] [20] established the empirical relationship between the radial displacement of the tunnel wall and the distance of the tunnel face by using the best fitting method, and the equation is as follows Equation (7):  
\[\lambda(x)\ =\ \bigg{[}1+\exp\bigg{(}\frac{-x}{1.1R}\bigg{)}\bigg{]}^{-1.7} \tag{7}\]  
The displacement release rate of this equation on the face of the tunnel is 0.31.