# Strain-Softening Analyses of a Circular Bore under the Influence of Axial Stress
## 4 Numerical Examples
### Numerical Analysis
#### 4.2.1 Axial Stress: \(q=30\) MPa  
For, \(q=30\) MPa, \(p_{c}=5.77\) MPa, when the internal pressure is less than \(p_{c}\), the plastic zone will develop in which \(\sigma_{\theta}>\sigma_{z}\ >\ \sigma_{r}\). While the internal pressure continues to decrease, two kinds of plastic zones form at certain stages. After the support pressure is completely released, the distribution of the stresses and displacement in the plastic zones is displayed in Figure 8. Both plastic zones are observed. In the outer plastic zone, \(\sigma_{\theta}>\sigma_{z}\ >\ \sigma_{r}\) and the normalized plastic radius \(R/r_{0}=1.754\). In the inner plastic zone, \(\sigma_{\theta}=\ \sigma_{z}>\ \sigma_{r}\). The normalized interface radius (\(\overline{R}/r_{0}\)) is 1.6. The dimensionless radial displacement at the bore wall is 4.35.  
\begin{table}
\begin{tabular}{l l} \hline
**Rock Parameter** & **Value** \\ \hline Radius of bore, \(r_{0}\) (m) & 3 \\ \hline Initial stress, \(\sigma_{0}\) (MPa) & 30 \\ \hline Out of plane stress, \(q\) (MPa) & 30 \\ \hline Internal pressure, \(p_{in}\) (MPa) & 0 \\ \hline [PERSON]’s modulus, \(E\) (GPa) & 27 \\ \hline [PERSON]’s ratio, \(\
u\) & 0.22 \\ \hline \(\sigma_{p}\) (MPa) & 1.5 \\ \hline \(\sigma_{p}\) (deg) & 50.9 \\ \hline \(\sigma_{p}\) (deg) & 15.5 \\ \hline \(\sigma_{r}\) (Mpa) & 0.7 \\ \hline \(\sigma_{r}\) (deg) & 39 \\ \hline \(\sigma_{r}\) (deg) & 7.5 \\ \hline \(\eta^{*}\) & 0.004 \\ \hline \end{tabular}
\end{table}
Table 1: Geometric and mechanical parameters.  
Figure 7: Convergence of dimensionless displacement in plastic zone.  
A contrastive analysis of the results presented in Figures 8-10 shows that the dimensionless radial displacement at the bore wall, the normalized plastic radius and the normalized interface radius obtained by the strain-softening model fall between the EBP and EP models.  
The normalized plastic radii (1.82, 1.754 and 1.29), normalized interface radii (1.71, 1.60 and 1.14) and dimensionless radial displacements (4.52, 4.35 and 1.64) correspond, respectively, to the EBP model, strain-softening model and EP model.  
When the geomaterial behaves in an EBP manner, the plastic region in the surrounding rock mass and the inward displacement at the bore wall are the largest. For EP media, the plastic region and inward displacement are the smallest.  
Two types of plastic zones (\(\sigma_{\theta}=\sigma_{z}\ >\ \sigma_{r}\) and \(\sigma_{\theta}>\sigma_{z}\ >\ \sigma_{r}\)) around the bore appear for the 3 models. Figure 11 shows the evolution of the radii of the elastic-plastic and softening-residual interfaces, denoted by \(R\) and \(R_{s}\), for different strain-softening slope. Figure 12 shows the corresponding GRCs.  
Figure 11: Evolution of (**a**) plastic radii \(R\) and (**b**) softening radii \(R_{s}\) (the radii between softening and residual zones) in the strain-softening media with different softening slopes when \(q=30\) MPa.  
It should be noted that \(\eta^{*}\) is a parameter related to plastic strain and that it controls the strain-softening slope, i.e., a different \(\eta^{*}\) means a different strain-softening slope. The larger the \(\eta^{*}\), the smaller the slope of the softening. When \(\eta^{*}\) is large enough, the slope of the softening is zero, and strain-softening model becomes the EP one. While it becomes very small, the slope of the softening is infinite (i.e., the peak strength changes into residual strength abruptly), and strain-softening model becomes EBP one.  
In Figures 11 and 12, the results are very close to that of the EBP model when \(\eta^{*}=0.002\). It can be inferred that the results are close to those of the EP model when \(\eta^{*}>0.016\).  
Figure 11b shows that part of the plastic region is in the residual strength stage and the other part is in the strain-softening stage when \(\eta^{*}=0.002\), \(0.004\) and \(0.008\). The strain-softening zone area becomes small as \(\eta^{*}\) decreases.