## 3 Theoretical Analysis  
Initially, the hydrostatic in situ stress (\(p_{0}\)) acts throughout soil mass around the tunnel/sinkhole. Then, with the tunnel excavation, the internal supporting pressure gradually decreases from \(p_{0}\) to \(\sigma_{T}\), and a plastic zone forms with an outer radius of \(c\) as shown in Figure 2. Under the axisymmetric conditions, the polar cylindrical coordinates (\(r\), \(\theta\), \(z\)) or the polar spherical coordinates (\(r\), \(\theta\), \(\phi\)) are adopted to account for particle position. Taking compression as positive, the stress boundary conditions of the cavity contraction problem are therefore defined as:  
\[\left.\sigma_{r}\right|_{r=b} =p_{0} \tag{2}\] \[\left.\sigma_{r}\right|_{r=a} =\sigma_{T}\] (3) \[\left.\sigma_{r}\right|_{r=c} =\sigma_{c} \tag{4}\]  
where \(\sigma_{r}\) is the radial stress component; \(r\) is the position of a soil particle; \(c\) denotes the radius of the elastic-plastic boundary; \(\sigma_{c}\) is the radial stress at the elastic-plastic boundary. Figure 1: Schematic of a tunnel heading and the cavity contraction model: (**a**) schematic of a tunnel heading; (**b**) the cavity contraction model. The stress components should be in equilibrium in the radial direction, which can be written as:  
\[\frac{\mathrm{d}\sigma_{r}}{\mathrm{d}r}+\frac{k}{r}(\sigma_{r}-\sigma_{\theta})=0 \tag{5}\]  
where \(\sigma_{\theta}\) is the circumferential stress; \(k\) is defined as the shape factor and \(k=1\) for the cylindrical cavity and \(k=2\) for the spherical cavity. As shown in Figure 2, in the plastic region (\(a<r<c\)), the soil behaves in an elastic-plastic manner. In the unloading process, \(\sigma_{r}\) is the minor principal stress while \(\sigma_{\theta}\) becomes the major principal stress. For soft clay in undrained conditions, the Tresca yielding function can be expressed as:  
\[\sigma_{\theta}-\sigma_{r}=2s_{\mathrm{u}} \tag{6}\]  
Combining Equations (5) and (6), the spatial derivative of the radial stress can be derived as:  
\[\frac{\mathrm{d}\sigma_{r}}{\mathrm{d}r}=\frac{2 ks_{\mathrm{u}}}{r} \tag{7}\]  
Integrating Equation (7) over the interval \([a,c]\), the internal supporting pressure that is the function of the elastic-plastic boundary can be written as:  
\[\sigma_{T}=\sigma_{c}-2 ks_{\mathrm{u}}\ln(c/a) \tag{8}\]  
In the elastic region (\(c<r<b\)), only elastic deformation occurs and the stress-strain relationship obeys [PERSON]'s law:  
\[\varepsilon_{r}=-\frac{\mathrm{d}u}{\mathrm{d}r}=\frac{1-v^{2}(2-k)}{E}\bigg{[} \sigma_{r}-\frac{kv}{1-v(2-k)}\sigma_{\theta}\bigg{]} \tag{9}\]  
\[\left.\sigma_{r}\right|_{r=b} =p_{0} \tag{2}\]0  
where \(\varepsilon_{r}\) and \(\varepsilon_{\theta}\) are the radial and circumferential strains, respectively; \(u\) is the displacement of a soil particle and is purely in the radial direction; \(E\) and \(v\) are the elastic modulus and [PERSON]'s ratio, respectively. Combining Equations (2), (4), (5), (9) and (10), the stress components in the elastic region can be obtained as:  
\[\left.\sigma_{r}\right|_{r=b} =p_{0} \tag{2}\]1  
\[\left.\sigma_{r}\right|_{r=b} =p_{0} \tag{2}\]2 Distribution of the elastic and plastic zone. At the elastic-plastic boundary (i.e., \(r=c\)), the stress components should also satisfy the yielding function (Equation (8)). Therefore, substituting Equations (11) and (12) into Equation (6), we can get:  
\[\left.\sigma_{r}\right|_{r=b} =p_{0} \tag{2}\]3  
Combining Equations (8) and (13), the relationship between the internal supporting pressure and the radius of the elastic-plastic boundary can be derived as:  
\[\left.\sigma_{r}\right|_{r=b} =p_{0} \tag{2}\]4  
It is worthwhile to note that the abovementioned analysis is conducted regarding the finite thickness of the cavity wall (named finite solution and hereafter). The tunnel is constructed, however, in semi-infinite space and the soil mass in the side and downward directions is infinite.