# Failure in the Tension Zone around a Circular Tunnel Excavated in Saturated Porous Rock
## 2 The H&B Criterion in the Compression and Tension Zone
### Determination of the H&B Parameters of a Porous Rock  
The Vosges sandstone is a porous, soft rock, classified as a pink quartz sandstone, with a few percent of feldspar and white mica [42]. This rock is poorly cemented, and the cohesion is due to the interpenetration between grains. The porosity is about 22%. The results of triaxial tests carried out on this sandstone show the occurrence of (1) a combination of axial splitting and inclined failure surfaces at low pressure (\(\approx\)0); (2) one or two parallel bands (failure surfaces) visible throughout the specimen with confinements between 10 and 30 MPa; (3) Conjugate shear bands with confinements from 40 to 60 MPa [42]. The considered range of confinement of the triaxial tests is 0.1-40 MPa because the transition from brittle to ductile failure occurs at 40 MPa. The tests are interpreted by assuming \(s=1\) and \(a=0.5\) (intact rock material, without crack damage). Figure 2a shows two typical interpretations of the laboratory tests with the H&B criterion, with the following ranges of confinement: (1) 0.1-40 MPa (2) 0.1-30 MPa. The data fitting clearly indicates a higher predicted value of \(\sigma_{t}\) when the confinement range is 0.1-40 MPa, as expected. As the tensile strength of Vosges sandstone is not measured, Equations (2) and (3) are used to estimate the uniaxial tensile strength of this rock. From the results of the tests carried out by [42], the estimated properties are: \(\sigma_{c_{-},exp}\approx 35\) MPa and \(\sigma_{ci}\cong 17\) MPa. This value of \(\sigma_{ci}\cong 48\%\sigma_{c_{-},exp}\) is in agreement with [2; 3; 4; 5; 6; 7; 8; 41]. By assuming \(\alpha=5\), the stress ratio is \(R=10.3\). The estimated tensile strength is \(\sigma_{t}=-3.4\) MPa, which is lower than the predicted tensile strength of Figure 2a. In the analysis of rock failure at the boundary of the tunnel, the confinement \(\sigma^{\prime}{}_{3}\) is expected to be lower (<5 MPa) than \(\sigma^{\prime}{}_{3}\) used in the high compression zone of triaxial tests. However, the available confinements in the compression zone are 0.1 MPa and 10 MPa. The red line in Figure 2b shows data fitting in the range \(-3.4\leq\sigma^{\prime}{}_{3}\leq 10\) MPa. The regression gives \(m_{t}=11.6\) and \(\sigma_{t}=-3.1\) MPa. For comparison, the figure also shows data fitting in the confinement range \(0.1\leq\sigma^{\prime}{}_{3}\leq 30\) MPa. By introducing \(m_{c}=7.8\), \(m_{t}=11.6\) and \(\sigma_{c_{-},exp}/\sigma_{ci}\cong 2.06\) in Equation (4), a back calculation of the constants \(\alpha\) and \(\beta\) gives \(\alpha=5.6\) and \(\beta=3.9\). These values are in agreement with the range proposed by [PERSON] [41]. The tensile strength calculated with the relationship \(\sigma_{c}/\sigma_{t}=0.81m+7\), proposed by [PERSON] and [PERSON] [38], with \(\sigma_{c}=42.2\) MPa and \(m=7.8\), gives \(\sigma_{t}=-3.2\) MPa, which is in agreement with the previous finding. Figure 2b also shows the H&B strength envelope with this cut-off (green line). Although the difference between the uniaxial tensile strengths from the regressions shown in Figure 2b is not so high (\(\sigma_{t}=-3.1\) MPa and \(\sigma_{t}=-5.3\) MPa), however, the extent of the tensile zones identified by the two data fitting can change the results of the analysis of rock failure near to the excavation boundary. The strength envelope with the H&B cut-off has a tension zone, which is intermediate between the two other cases shown in Figure 2b. The parameters of the three data fitting shown in Figure 2 (Case 1, 2, 3, and 4) are used to carry out sensitivity analysis of rock failure close to the boundary of the excavation (in the tension zone). Here the effect of uniaxial compressive strength reduction observed in saturated specimens is disregarded, because the purpose is a comparative analysis related to the data fitting procedure. The strength parameters of Case 1 are the less reliable for the tension zone and their use has just the purpose to complete the scenario of this study.