# Failure in the Tension Zone around a Circular Tunnel Excavated in Saturated Porous Rock
## 4 Rock Failure in the Tension Zone during Tunnel Excavation: Results and Discussion  
The pattern of rock failure in the tension zone in undrained condition (\(u>0\) with \(\Delta u=0\) or \(\Delta u\
eq 0\)), can be analyzed with the Mohr circles and the origin of planes. Figure 4 shows two conjugate failure planes that isolate a prismatic rock solid. The shape of the rock-solid changes in relation to the azimuth, and induced state of stress and pore pressure. A similar shape of damage in the tension zone of a circular test tunnel was noted and identified with AE, even though no cracking or fracturing could be visually observed [4]. On the other hand, the well-observed and investigated brittle failure process results in classic V-shaped notches, in the region of maximum compressive stress. In this case, shallow spalling notches, experienced in tunnels in the compression zone, are associated with pure tensile failure at the microscale and cause thin slivers or plates of rock to peel off the tunnel surface [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50]. However, under overall tensile loading, the stress levels of crack initiation (\(\sigma_{c}\))\({}_{t}\) and crack propagation (\(\sigma_{cd}\))\({}_{t}\) (stress level at which crack coalescence occurs) are very close to the tensile peak strength \(\sigma_{t}\): (\(\sigma_{c}\))\({}_{t}\)\(\cong\) (\(\sigma_{cd}\))\({}_{t}\)\(\cong\)\(\sigma_{t}\)[41]. The occurrence of failure in the tension zone results in a negative radial effective stress, induced by the pore pressure. In this case, failure is expected to occur at the time of the excavation of the tunnel, because the induced tensile stress is close to \(\sigma_{t}\). The variation of \(u_{lim}\) with the state of stress \(S\) and the different couples of \(m\) and \(\sigma_{c}\) (Figure 2 and Table 1), with \(P_{o}=0\) and \(X<1\), is analyzed in the following. Two cases of far field stresses, which induce different \(S\) at \(r=R\), are selected (Table 2). The state of stress \(S\) = 8 MPa occurs in case A and Case B at \(\theta=90^{\circ}\) and at \(\theta=0^{\circ}\), respectively. Figure 4: Failure in the tension zone as a function of the induced state of stress. The dotted red lines are the failure planes. The first analyses are in undrained condition with \(\Delta u=0\), which results in a total pore pressure equal to the initial pore pressure \(u=u_{imi}\) at \(r\geq R\). Figure 5a shows that \(u_{lim}\) increases when \(S\) decreases, according to Equation (10). The figure shows that, due to the change of \(m\) and \(\sigma_{c}\) (high \(m\) combined with low \(\sigma_{c}\) and viceversa), \(u_{lim}\) is almost equal in Cases A (S = 8 MPa) and B (S = 4 MPa). Figure 5a also shows that \(u_{lim}\) decreases with increasing \(m\). This outcome is related to the tension zone, which is reduced when the slope of the strength envelope increases. However, here any \(m\) is associated with a different \(\sigma_{c}\) and the term \(X\) varies along the lines. Figure 5b shows the influence of the strength parameters \(\sigma_{c}\) and \(m\) when \(S=16\) MPa at \(\theta=0^{\circ}\) (Case A). The figure shows the variation of \(u_{lim}\) with \(m\) (\(5.6\div 11.6\)) at three fixed uniaxial compressive strengths: \(\sigma_{c}=36.6\) MPa \(\sigma_{c}=42.2\) MPa \(\sigma_{c}=48.5\) MPa. A decrease of the influence of \(m\) in the determination of \(u_{lim}\) occurs at the lower \(\sigma_{c}\). However, the effect of \(\sigma_{c}\) is related to the magnitude of \(S\). In fact, an increase of \(X\), due to a lower \(\sigma_{c}\), corresponds to a decrease of the influence of \(m\) in the determination of \(u_{lim}\), because \(\sigma_{c}\) becomes closer to \(S\). The green symbol in Figure 5b refers to \(X=1\), which corresponds to \(u_{lim}=0\). The role of \(X\) is furtherly investigated in Figure 6a, with the strength parameters of Case 2 and Case 3 (Table 1).