# Strain-Softening Analyses of a Circular Bore under the Influence of Axial Stress
## 4 Numerical Examples
### Numerical Analysis
#### 4.2.3 Influences of the Dilation Angle on Numerical Results  
In plasticity theory, either an associative flow rule or a non-associative flow rule is used to derive the incremental plastic strain. When the associative flow rule is used, the plastic potential function is exactly the same as the yield function and the dilation angle equals the internal friction angle in the M-C criterion, whereas, in the non-associative flow rule, the plastic potential function is different from the yield function.  
It is an observable fact that dilatancy is dependent on shear plastic strain [2; 22]. Moreover, the dilation angle is not equal to the internal friction angle. According to [PERSON] and [PERSON] [22], the following relationships are recommended, and the dilation angle and the friction angle are in the range \(\psi=\varphi/4\) for good-quality rock and \(\psi=0\) for poor-quality rock. This means that the dilation angle is much smaller than the internal friction angle and that the non-associative flow rule is more appropriate for elasto plastic analysis.  
Figure 18 shows GRC in the strain-softening media with \(\eta^{*}=0.004\) and \(q=30\) MPa. Different evolution laws of dilation angle are considered.  
First, the associative flow rule was employed, and \(\psi p=\varphi p=50.9^{\circ}\) and \(\psi r=\varphi r=39^{\circ}\). The dimensionless radial displacement increased as the internal pressure decreased. The dimensionless radial displacement was 31.8 when the internal pressure was completely released.  
\begin{table}
\begin{tabular}{c c c c} \hline  & \(\frac{R}{\tau_{0}}\) & \(\frac{-}{R_{0}}\) & \(2\ u\ G/(\sigma_{0}\tau_{0})\) \\ \hline \(q=30\) MPa & 1.754 & 1.60 & 4.35 \\ \hline \(q=60\) MPa & 1.85 & 1.85 & 5.59 \\ \hline \end{tabular}
\end{table}
Table 2: Comparison of results for different out-of-plane stress.  
Figure 17: GRC in the strain-softening media with different softening slopes when \(q=60\) MPa.  
When \(\psi p=1/2\), \(\varphi p=25.45^{\circ}\), \(\psi r=1/2\), and \(\varphi r=19.5^{\circ}\), the flow rule was non-associative. The dimensionless radial displacement was 8.3 when the internal pressure was zero. When \(\psi p=15.5^{\circ}\) and \(\psi r=7.5^{\circ}\), the dimensionless radial displacement is 5.4. GRC was also presented in Figure 18 for comparison. In this case, the flow rule was also non-associative.  
Figure 19 shows GRC in the strain-softening media with \(\eta^{*}=0.004\) and \(q=60\) MPa, and similar analyses were carried out. When the associative flow rule was used, the dimensionless radial displacement was 59.5. When the non-associative flow rule was employed, the dimensionless radial displacement was 11.7 for \(\psi p=1/2\)\(\varphi p\) and \(\psi r=1/2\)\(\varphi r\), and 7.2 for \(\psi p=15.5^{\circ}\) and \(\psi r=7.5^{\circ}\).  
The dimensionless radial displacements are summarized in Table 3 for different out-of-plane stresses. The results indicated that the displacements obtained by the associative flow rule were larger than those by the non-associative flow rule. When the dilation angle was much smaller than the internal friction angle, the difference was huge.  
Figure 19: GRC in the strain-softening media with different evolution of dilation angle when. \(q=60\) MPa.  
Figure 18: GRC in the strain-softening media with different evolution of dilation angle when \(q=30\) MPa.  
When the out-of-plane stress increases, the difference between the displacements obtained by associative flow rule and those by non-associative flow rule also increases.