## 5 Verification, Comparison, and Discussion  
### Verification for Finite Strain Procedure  
For the verification, this study compares the proposed solution with the numerical solutions in [37; 38]. The parameters of the MC rock mass listed in Table 1 are the same as those in 'Figure 6' provided by Ref. [37]. The relations between the dimensionless  
Figure 6: Calculation flowchart for elastic-softening-residue state.  
displacement \(u_{0}/R_{0}\), \(R_{\rm P}/R_{0}\), \(R_{\rm r}/R_{0}\) and the dimensionless supporting pressure \(p/P_{0}\) are visualized by the scattered diagrams in Figure 7 with two sets of data calculated by the procedure in this paper and former studies. Ref. [37] presents the results solved by the recursive program ([PERSON]'s solution) and obtained by the finite finite-difference method (FDM).  
By the proposed procedure in this paper, the calculated limit support pressures \(p_{\rm TP}^{*}\) and \(p_{\rm pe}^{*}\) are equal to 0.0945 MPa and 0.2006 MPa respectively. As shown in Figure 7, the GRC and the evolutions of the softening region radius and residual region radius calculated by the proposed procedure are virtually identical with those of [PERSON]'s solution, which verifies the accuracy of the proposed numerical finite strain procedure. Without the supporting pressure, \(u_{0}/R_{0}\), \(R_{\rm P}/R_{0}\), and \(R_{\rm r}/R_{0}\) are 0.3196, 1.884, and 2.260 respectively, slightly different with 0.3235, 1.87 and 2.25 in [PERSON]'s study. This difference is caused mainly by the approximant treatment of the infinity.  
In [PERSON]'s study, \(k_{\infty}\) is taken as 50 to approximate the infinite boundary, whereas \(k_{\infty}\) equals \(10^{5}\) in this section. As shown in Figure 8, the value of \(k_{\infty}\) obviously affects the computational accuracy of \(u_{0}/R_{0}\), \(R_{\rm P}/R_{0}\) and \(R_{\rm r}/R_{0}\). When \(k_{\infty}\) reaches 1000, the calculation results stabilize. In addition, the iteration times \(n\) have a decisive influence on the accuracy as well. As shown in Figure 9, with the number of iterations increasing, the values of \(u_{0}/R_{0}\), \(R_{\rm P}/R_{0}\) and \(R_{\rm r}/R_{0}\) gradually converge and the magnitude of the calculation error \(|\alpha|\) decreases to about \(10^{-14}\) which reaches the highest accuracy when \(k_{\infty}=10^{7}\). After iterating only 10 times, \(u_{0}/R_{0}\), \(R_{\rm P}/R_{0}\) and \(R_{\rm r}/R_{0}\) become rather stable. When the tolerances of the calculation relative error \({\rm Tol}(\alpha)\) are \(10^{-6}\), \(10^{-8}\), \(10^{-10}\), \(10^{-12}\)  
\begin{table}
\begin{tabular}{c c} \hline
**MC Rock** & **Value** \\ \hline Radius of opening, \(R_{0}\) (m) & 3 \\ Initial stress, \(P_{0}\) (MPa) & 1 \\ Young’s modulus, \(E\) (MPa) & 30 \\ Poisson ratio, \(v\) (-) & 0.3 \\ Critical plastic shear strain, \(\gamma\)P\({}^{*}\) (-) & 0.15 \\ \(c_{\rm p}\) (MPa) & 0.2 \\ \(c_{\rm r}\) (MPa) & 0.02 \\ \(\varphi_{\rm P}\) (Deg) & 40 \\ \(\varphi_{\rm r}\) (Deg) & 20 \\ \(\psi_{\rm P}\) (Deg) & 10 \\ \(\psi_{\rm r}\) (Deg) & 5 \\ \hline \end{tabular}
\end{table}
Table 1: Geometry and material property parameters.  
Figure 7: Comparison between the results of this paper and [PERSON]’s: (**a**) Ground response curve (GRC); (**b**) radii of residue region and softening region.  
and \(10^{-14}\), the time for one whole calculation is 0.89 s, 1.18 s, 1.72 s, 1.98 s, and 2.82 s in this numerical example. It shows that for the proposed procedure the significant improvement of the precision does not affect the calculation efficiency excessively.