## 3 Cylindrical Cavity Expansion in the SCC Soil
### Elastoplastic Analysis
According to Equation (11), the term \(\partial p^{\prime}/\partial e^{p}_{v}\) in Equation (25) can be obtained as follows:  
\[\frac{\partial p^{\prime}}{\partial e^{p}_{v}}=-\frac{(1+e)p^{\prime}}{\kappa} \tag{26}\]  
By combining Equation (22) and the non-associated flow rules Equations (5), (6), or (8), the plastic deviator strain increment \(d\varepsilon^{p}_{d}\) can be expressed in the following unified form:  
\[d\varepsilon^{p}_{d}=\frac{\beta}{K_{p}}\left\{\frac{\partial f}{\partial \sigma^{\prime}_{i}}\right\}^{T}\left\{d\sigma^{\prime}_{i}\right\}=\Lambda \frac{\partial g}{\partial q} \tag{27}\]  
where \(\beta=\partial e^{p}_{d}/\partial e^{p}_{v}\) represents the plastic flow rule, which follows the different forms in the processes of hardening (\(\beta=\beta_{1}\)), softening (\(\beta=\beta_{2}\)), and crushing (\(\beta=\beta_{3}\)). Combining Equations (22) and (27), the components of the plastic strain increments can be obtained as follows:  
\[ \begin{array}{ll}de^{p}_{i}&=\Lambda\frac{\partial g}{\partial\sigma^{\prime }_{i}}=\Lambda\frac{\partial g}{\partial p^{\prime}}\frac{\partial p^{\prime}} {\partial\sigma^{\prime}_{i}}+\Lambda\frac{\partial g}{\partial q}\frac{ \partial q}{\partial\sigma^{\prime}_{i}}\\ &=\frac{1}{K_{p}}\left\{\frac{\partial p^{\prime}}{\partial\sigma^{\prime}_{i} }+\beta\frac{\partial q}{\partial\sigma^{\prime}_{i}}\right\}\left\{\frac{ \partial f}{\partial\sigma^{\prime}_{j}}\right\}^{T}\left\{d\sigma^{\prime}_{j} \right\}\end{array} \quad i,\ j\ =\ r,\ \theta,\ z \tag{28}\]  
Rewriting Equation (28) in matrix form, the following can be obtained:  
\[\left\{ \begin{array}{ll}d\varepsilon^{p}_{r}\\ d\varepsilon^{p}_{g}\\ d\varepsilon^{p}_{z}\end{array} \right\}=\frac{1}{K_{p}}\left[ \begin{array}{ ccc}A_{r}B_{r}&A_{r}B_{g}&A_{r}B_{z}\\ A_{\theta}B_{r}&A_{\theta}B_{g}&A_{\theta}B_{z}\\ A_{z}B_{r}&A_{z}B_{\theta}&A_{z}B_{z}\end{array} \right]\left\{ \begin{array}{ ll}d\sigma^{\prime}_{r}\\ d\sigma^{\prime}_{g}\\ d\sigma^{\prime}_{z}\end{array} \right\} \tag{29}\]  
where  
\[A_{i}=\frac{\partial p^{\prime}}{\partial\sigma^{\prime}_{i}}+\beta\frac{ \partial q}{\partial\sigma^{\prime}_{i}}=\frac{1}{3}+\frac{3\beta\left(\sigma^{ \prime}_{i}-p^{\prime}\right)}{2q} \tag{30}\]\[B_{j}=\frac{\partial f}{\partial\sigma_{j}^{\prime}}=\frac{p^{\prime}\big{(}M^{2}- \eta^{2}\big{)}}{3}+3\big{(}\sigma_{i}^{\prime}-p^{\prime}\big{)} \tag{31}\]  
Substituting Equations (11) and (29) into Equation (18), the relationship between total strain components and effective stress components can be expressed as follows:  
\[\left\{ \begin{array}{l}d\varepsilon_{r}\\ d\varepsilon_{\theta}\\ d\varepsilon_{z}\end{array} \right\}=\left[ \begin{array}{cccc}\frac{A_{r}B_ {k}}{K_{p}}+\frac{1}{E}&\frac{A_{r}B_{\theta}}{K_{p}}-\frac{v}{E}&\frac{A_{r} B_{\theta}}{K_{p}}-\frac{v}{E}\\ \frac{A_{r}B_{r}}{K_{p}}-\frac{v}{E}&\frac{A_{r}B_{\theta}}{K_{p}}+\frac{1}{E} &\frac{A_{r}B_{\theta}}{K_{p}}-\frac{v}{E}\\ \frac{A_{r}B_{r}}{K_{p}}-\frac{v}{E}&\frac{A_{r}B_{\theta}}{K_{p}}-\frac{v}{E} &\frac{A_{r}B_{r}}{K_{p}}+\frac{1}{E}\end{array} \right]\left\{ \begin{array}[] {l}d\sigma_{r}^{\prime}\\ d\sigma_{\theta}^{\prime}\\ d\sigma_{z}^{\prime}\end{array} \right\} \tag{32}\]  
By taking the inverse form of Equation (32), the components of stress increment can be expressed in terms of the components of strain increment as  
\[\left\{ \begin{array}{l}d\sigma_{r}^{\prime}\\ d\sigma_{\theta}^{\prime}\\ d\sigma_{z}^{\prime}\end{array} \right\}=\frac{1}{\Gamma}\left[ \begin{array}[] {cccc}C_{rr}&C_{\theta}&C_{rz}\\ C_{\theta r}&C_{\theta\theta}&C_{\theta z}\\ C_{zr}&C_{z\theta}&C_{zz}\end{array} \right]\left\{ \begin{array}{l}d \varepsilon_{r}\\ d\varepsilon_{\theta}\\ d\varepsilon_{z}\end{array} \right\} \tag{33}\]  
where  
\[C_{ii}=E\Big{[}K_{p}(1-v^{2})+E(A_{j}B_{j}+A_{k}B_{k})+Ev(A_{j}B_{k}+A_{k}B_{ j})\Big{]}\ \ i,j,k\ =r,\theta,Z;\ i\ \
eq\ j\ \
eq\ k \tag{34a}\] \[C_{ij}=E\Big{[}K_{p}(v+v^{2})-EA_{i}B_{j}-Ev(A_{i}B_{k}+A_{k}B_{j}-A_{k}B_{k}) \Big{]}\ \ i,j,k=r,\theta,z;\ i\ \
eq\ j\ \
eq\ k\] (34b) \[\Gamma=(v+1)\Big{[}