# 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  
The Hoek and Brown criterion [27] was introduced in 1980 and was derived for applications in the underground excavation in hard brittle rocks. Practitioners in rock engineering have applied the Hoek and Brown criterion successfully for over 40 years to a wide range of intact and fractured rock types [36; 37]. The original criterion has been updated by the authors based on the experience acquired in its practical application in several geological and engineering contexts [38]. For a saturated rock with a water pore pressure \(u\) the criterion is expressed in terms of effective stresses: \({\sigma^{\prime}}_{1}={\sigma}_{1}-u\) and \({\sigma^{\prime}}_{3}={\sigma}_{3}-u\). The general nonlinear criterion for intact rock is:  
\[{\sigma^{\prime}}_{1}={\sigma^{\prime}}_{3}+\left(m\sigma_{c}{\sigma^{\prime }}_{3}+s\sigma_{c}^{2}\right)^{a}, \tag{1}\]  
where: \(\sigma_{c}\) is the uniaxial compressive strength of the intact rock obtained from data fitting, and \(m\) and \(s\) are empirical dimensionless constants. For hard intact rock, the parameter \(s\) is always equal to 1, and the constant \(a=0.5\). The uniaxial compressive strength \(\sigma_{c}\) and the constant \(m\) are obtained by data fitting of uniaxial and triaxial compression tests. The range of applicability of the criterion is determined by the transition from shear to ductile failure. Furthermore, tensile failure is not dealt with by the H&B criterion [38]. As tensile failure can occur in some rock engineering problems, [PERSON] [39] proposed the Griffith crack theory to set the ratio of compressive strength \(\sigma_{c}\) to tensile strength \(\sigma_{t}\). For practical purposes, [PERSON] [39] proposed a tensile cut-off. [PERSON] and [PERSON] [38] suggested an empirical relationship, \(\sigma_{c}/\sigma_{t}=0.81\ m+7\), for the definition of the cut-off. The parameter \(m\) is a constant, which describes the slope of the strength envelope. The value of \(m\) affects the strength in the compression zone (high and low confinement) and in the tension zone (Figure 1b). In general, the uniaxial compressive strength estimated with data fitting with H&B criterion can be higher from the measured one, in particular when data fitting is carried out with the use of high confinements. On the other hand, the tensile strength obtained with data fitting is underestimated for strong and brittle rocks and overestimated for soft rocks [40]. An analysis of data fitting carried out by [PERSON] [41], indicated that the H&B criterion \"is very good for hard, strong crystalline rocks at high confining pressure, but much less so in the low confining pressure and tension zone\", consequently the \(m\) value is not a constant but confining pressure dependent. Figure 1b clearly shows that the H&B criterion predicts a higher tensile strength with low values of \(m\). [PERSON] [41] presented a study focused on the practical estimates of the tensile strength with the H&B criterion in weak rocks with pores. The study indicates that the compressive stress to tensile stress ratio at crack initiation is less than 8 (according to the [PERSON] theory). Using the same analog of [PERSON]'s theory and accounting for the difference of crack propagation in compression and tension, the following stress ratio \(R\) is proposed:  
\[R=\alpha\frac{\sigma_{c_{-},\exp}}{\sigma_{ci}}, \tag{2}\]  
where: \(\sigma_{c_{-},exp}\) is the experimental value of the uniaxial compressive strength, \(\sigma_{ci}\) is the crack initiation stress level, and \(\alpha\) is a constant for the tension zone.