## 3 Cylindrical Cavity Expansion in the SCC Soil
### Elastoplastic Analysis
\begin{array}{cccc}K_{p}(1-v-2v^{2})+E(1-v)(A_{r}B_{r}+A_ {\theta}B_{\theta}+A_{z}B_{z})\\ +Ev(A_{r}B_{\theta}+A_{\theta}B_{r}+A_{\theta}B_{z}+A_{z}B_{\theta}+A_{r}B_{z}+ A_{z}B_{r})\end{array} \Big{]}\ \ i,j,k=r,\theta,z;\ i\ \
eq\ j\ \
eq\ k \tag{34c}\]  
Under the undrained and plane-strain assumptions, the volumetric strain increment \(d\varepsilon_{v}\) and vertical strain increment \(d\varepsilon_{z}\) are zero, which leads to \(d\varepsilon_{r}=-d\varepsilon_{\theta}\), and therefore Equation (33) becomes  
\[d\sigma_{r}^{\prime} =\frac{-C_{rr}+C_{r\theta}}{\Gamma}d\varepsilon_{\theta} \tag{35a}\] \[d\sigma_{\theta}^{\prime} =\frac{-C_{\theta r}+C_{\theta\theta}}{\Gamma}d\varepsilon_{\theta}\] (35b) \[d\sigma_{z}^{\prime} =\frac{-C_{z\tau}+C_{z\theta}}{\Gamma}d\varepsilon_{\theta} \tag{35c}\]  
According to the large strain theory, the tangential strain increment \(d\varepsilon_{\theta}\) in Equation (35a-c) can be expressed in the logarithmic strain form as  
\[d\varepsilon_{\theta}=-\frac{dr}{r} \tag{36}\]  
Substituting Equation (36) into Equation (35a-c), it can be seen that three stress components, \(\sigma^{\prime}{}_{r},\sigma^{\prime}{}_{\theta}\), and \(\sigma^{\prime}{}_{z}\), are functions of radial distance \(r\). Note that the expressions of \(C_{ij}\) and \(\Gamma\) in Equation (35a-c) is inconsistent in the hardening/softening and crushing processes. For the hardening/softening process of the structured soils, the variations of the stress components can be calculated by taking the values of \(\sigma^{\prime}\),\(r_{\left(xp\right)}\), \(\sigma^{\prime}{}_{\theta}\)(\(r_{xp}\)), and \(\sigma^{\prime}{}_{z}\)(\(r_{xp}\)) as the initial values. If the soil element attains the critical state but with \(\Delta e\
eq c\), the crushing of the structure will occur at a subsequent stage, which needs to be further solved by taking the values of \(\sigma^{\prime}{}_{r}\)(\(r_{xc}\)), \(\sigma^{\prime}{}_{\theta}\)(\(r_{xc}\)), and \(\sigma^{\prime}{}_{z}\)(\(r_{xc}\)) as the initial values. However, it is complex to determine \(r_{xc}\) because it depends not only on the current radial location \(r\), but also on the nonlinear displacement \(U_{r}\) during the hardening/softening process. To simplify the solution procedure, the auxiliary variable \(\xi\), defined by Chen and Abousleiman [23] as \(U_{r}/r\), is skillfully introduced, where \(U_{r}=r-r_{0}\) is the radial displacement for a specific soil element. This auxiliary variable takes the incremental form as follows:  
\[\frac{d\xi}{1-\xi}=\frac{dr}{r} \tag{37}\]  
Combining Equations (36) and (37), it can be seen that the tangential strain increment \(d\varepsilon_{\theta}\) is merely related to \(\xi\); therefore, Equation (35a-c) can be written as  
\[\frac{d\sigma^{\prime}_{r}}{d\xi} =\frac{C_{rr}-C_{r\theta}}{\Gamma(1-\xi)} \tag{38a}\] \[\frac{d\sigma^{\prime}_{\theta}}{d\xi} =\frac{C_{\theta r}-C_{\theta\theta}}{\Gamma(1-\xi)}\] (38b) \[\frac{d\sigma^{\prime}_{z}}{d\xi} =\frac{C_{zr}-C_{z\theta}}{\Gamma(1-\xi)} \tag{38c}\]  
Equation (38a-c) formulate the problem as a system of first-order differential equations with respect to a single variable \(\xi\), which are valid for any soil element around the cavity. The values of \(\sigma^{\prime}\), \(\sigma^{\prime}{}_{\theta}\), and \(\sigma^{\prime}{}_{z}\) at the instant of \(\xi=\xi_{p}\) and \(\xi=\xi_{c}\) are required as the initial values of the hardening/softening and crushing processes, where \(\xi_{p}\) and \(\xi_{c}\) correspond to the values of \(u_{r}/r\) at the positions \(r_{xp}\) and \(r_{xx}\), respectively. The values of \(\sigma^{\prime}{}_{r}(\xi_{c})\), \(\sigma^{\prime}{}_{\theta}(\xi_{c})\), and \(\sigma^{\prime}{}_{z}(\xi_{c})\) as the initial values of the crushing process can be determined from the hardening/softening process, owing to the continuity of deformation; therefore, the values of \(\sigma^{\prime}{}_{r}(\xi_{p})\), \(\sigma^{\prime}{}_{\theta}(\xi_{p})\), and \(\sigma^{\prime}{}_{z}(\xi_{p})\) are sufficient to solve the stress variation for a specific soil element.