## Appendix A  
With the small deformation hypothesis, the formulas for stress and displacement of rock mass in three regions are derived and listed as follows. All formula symbols are consistent with the above formulas.  
(1) Solutions in residue region  
The solution of the radial stress and the tangential stress in residue region can be expressed as Equations (A1) and (A2).  
\[\sigma_{\rm r}=\frac{\sigma_{0}(k_{\rm r}-1)+\sigma_{\rm c}}{k_{\rm r}-1}\bigg{(} \frac{r}{R_{0}}\bigg{)}^{k_{\rm r}-1}-\frac{\sigma_{\rm c}}{k_{\rm r}-1}\] (A1)  
\[\sigma_{\rm\theta}=\frac{k_{\rm r}[\sigma_{0}(k_{\rm r}-1)+\sigma_{\rm c}]}{k_ {\rm r}-1}\bigg{(}\frac{r}{R_{0}}\bigg{)}^{k_{\rm r}-1}-\frac{\sigma_{\rm c}} {k_{\rm r}-1}\] (A2)  
where \(k_{r}=\frac{1+\sin\varphi_{\rm r}}{1-\sin\varphi_{\rm r}}\) and \(\sigma_{\rm c}(\gamma^{p})=\frac{2\tau_{\rm c}\cos\varphi_{\rm r}}{1-\sin \varphi_{\rm r}}\). The solution of the displacement in the residue region is expressed as Equation (A3).  
\[u=C\frac{r^{k_{\rm r}}-r^{-K}{R_{0}}^{k_{\rm r}+K}}{K+k_{\rm r}}+D\frac{r-r^{- K}{R_{0}}^{K+1}}{K+1}+r^{-K}{R_{0}}^{K}u_{0}\] (A3)  
where \(C=\frac{1}{2G}\frac{\sigma_{0}(k_{\rm r}-1)+\sigma_{\rm c}}{(k_{\rm r}-1){R_ {0}}^{k_{\rm r}+1}}[1-v-Kv+k_{\rm r}(K-Kv-v)]\), \(D=-\frac{(K+1)(1-2v)}{2G}\Big{(}\frac{\sigma_{\rm c}}{k_{\rm r}-1}+P_{0}\Big{)}\) and \(K=\frac{1+\sin\varphi_{\rm r}}{1-\sin\varphi_{\rm r}}\).  
(2) Solutions in softening region  
The differential equations of the radial stress and of the plastic shear strain with respect to the original radius are derived as Equations (A4) and (A5).  
\[\frac{\mathrm{d}\sigma_{\rm r}}{\mathrm{d}r}=\frac{1}{r}\big{[}(k_{\rm p}( \gamma^{\rm p})-1)\sigma_{\rm r}+\sigma_{\rm c}(\gamma^{\rm p})\big{]}\] (A4)  
\[\frac{\mathrm{d}\gamma^{\rm p}}{\mathrm{d}r}=\frac{-\frac{1}{2G}\big{[}(1-v)k_ {\rm p}(\gamma^{\rm p})-v\big{]}\frac{\mathrm{d}\sigma_{\rm r}}{\mathrm{d}r}- \frac{1}{r}\frac{1}{2G}\big{\{}\big{[}k_{\rm p}(\gamma^{\rm p})-1\big{]} \sigma_{\rm r}+\sigma_{\rm c}(\gamma^{\rm p})\big{\}}-\frac{\gamma^{\rm p}}{r} }{\frac{1}{2G}\Big{[}(1-v)\sigma_{\rm r}\frac{\mathrm{d}k_{\rm p}(\gamma^{ \rm p})}{\mathrm{d}\gamma^{\rm p}}+(1-v)\frac{\mathrm{d}\sigma_{\rm c}(\gamma^ {\rm p})}{\mathrm{d}\gamma^{\rm p}}\Big{]}+\frac{\beta\bigg{[}\frac{\gamma^{ \rm p}}{1+K(\gamma^{\rm p})}\bigg{]}}{\frac{\gamma^{\rm p}}{\mathrm{d}\gamma^ {\rm p}}}}\] (A5)  
The displacement of rock mass material points in the softening region can be calculated by Equation (A6).  
\[u=\frac{r}{2G}\big{[}(1-v)(\sigma_{\rm\theta}-P_{0})-v(\sigma_{\rm r}-P_{0}) \big{]}+r\frac{\gamma^{\rm p}}{1+K(\gamma^{\rm p})}\] (A6)  
(3) Solutions in elastic region  
The solution of the radial stress and the tangential stress in the elastic region can be expressed as Equations (A7) and (A8).  
\[\sigma_{\rm r}=\frac{\sigma_{\rm\theta}^{R_{\rm p}}+\sigma_{\rm r}^{R_{\rm p}} }{2}-\frac{\sigma_{\rm\theta}^{R_{\rm p}}-\sigma_{\rm r}^{R_{\rm p}}}{2}\frac{{ R_{\rm p}}^{2}}{r^{2}}\] (A7)  
\[\sigma_{\rm\theta}=\frac{\sigma_{\rm\theta}^{R_{\rm p}}+\sigma_{\rm r}^{R_{\rm p }}}{2}+\frac{\sigma_{\rm\theta}^{R_{\rm p}}-\sigma_{\rm r}^{R_{\rm p}}}{2}\frac{{ R_{\rm p}}^{2}}{r^{2}}\] (A8)  
where \(\sigma_{\rm\theta}^{R_{\rm p}}\) and \(\sigma_{\rm r}^{R_{\rm p}}\) are the radial stress and tangential stress on the boundary between the elastic region and the plastic region (\(R_{p}\)).  
The displacement of rock mass in the elastic region can be calculated by Equation (A9).  
\[u=\frac{r}{2G}\big{[}(1-v)(\sigma_{\rm\theta}-P_{0})-v(\sigma_{\rm r}-P_{0}) \big{]}\] (A9)