# Strain-Softening Analyses of a Circular Bore under the Influence of Axial Stress
## 4 Numerical Examples
### Numerical Analysis
#### 4.2.2 Out-of-Plane Stress: \(q=60\) MPa  
Here the geometric and mechanical parameters presented in Table 1 is used except that \(q=60\) MPa.  
While the internal pressure is less than \(p_{c}\) (=6.5 MPa), the plastic zone first develops when \(\sigma_{z}>\sigma_{\theta}~{}>~{}\sigma_{r}\). Only one plastic zone with \(\sigma_{\theta}=~{}\sigma_{z}>~{}\sigma_{r}\) is formed in this case as the support pressure is completely released. The distribution of the stresses and displacement in the plastic zones is shown in Figure 13. The normalized plastic radius \(R/r_{0}=1.85\) and the dimensionless radial displacement at the bore wall is 5.59.  
Figure 12: GRC in the strain-softening media with different softening slopes when \(q=30\) MPa.  
Figure 13: (**a**) Normalized radial displacement, (**b**) normalized radial, tangential and axial stress in the surrounding rock mass in strain-softening model when \(q=60\) MPa and \(p_{in}=0.0\) MPa.  
Figures 14 and 15 show the distributions of the stresses and displacement in the plastic zones, which are obtained by means of EBP model and EP model.  
A comparison of the results presented in Figures 13-15 indicates that the dimensionless radial displacement at the bore wall and the normalized plastic radius obtained by the strain-softening model fall between those of the EBP and EP models.  
The normalized plastic radii are 1.88, 1.85 and 1.31, and the dimensionless radial displacements are 5.6, 5.59 and 1.82, corresponding to the EBP model, strain-softening model and EP model, respectively. For these 3 models, a similar phenomenon is observed concerning the magnitude of the normalized plastic radius and the dimensionless radial displacement at the bore wall.  
Two types of plastic zone (\(\sigma_{\theta}=\sigma_{z}\ >\ \sigma_{r}\) and \(\sigma_{\theta}>\sigma_{z}\ >\ \sigma_{r}\)) around the bore appear only for the EP model. Figure 16 shows the evolution of the radii of the elastic-plastic and softening-residual interfaces for different strain-softening slopes. Figure 17 shows the corresponding GRCs.  
In Figures 16 and 17, the results are very close to those of the EBP model when \(\eta^{*}=0.004\). However, we have the same curves as those obtained by EBP model if \(\eta^{*}=0.002\). This is a little different from the results shown in Figures 11 and 12 when \(q=30\) MPa and \(\eta^{*}=0.002\). It looks as though the influence of the out-of-plane stress is great.  
Figure 14: (**a**) Normalized radial displacement, (**b**) normalized radial, tangential and axial stress in the surrounding rock mass in EBP model when \(q=60\) MPa and \(p_{in}=0.0\) MPa.  
Figure 15: (**a**) Normalized radial displacement, (**b**) normalized radial, tangential and axial stress in the surrounding rock mass in EP model when \(q=60\) MPa and \(p_{in}=0.0\) MPa.  
Figure 16b shows that part of the plastic region is in the residual strength stage and the other part is in the strain-softening stage when \(\eta^{*}\) = 0.004 and 0.008.  
Figure 16: 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 \(q\) = 60 MPa.