## 2 Materials and Methods  
### Fundamentals of Brittle Failure Numerical Modelling  
Modelling rock mass behaviour presents several challenges; among these, the modelling of failure process, comprising nucleation, growth, and coalescence of microcracks, followed by strain localisation and macroscopic fracturing, is one of the toughest. Furthermore, the rock mass fabric, with the presence of pre-existing discontinuities such as joints, faults, bedding planes, etc., affects the failure process of intact rock in several ways. A straightforward way to predict the depth/extent of brittle failure would be to perform an elastic stress analysis and compare the induced stresses with the rock mass strength. However, with this method, the actual failed zone is significantly underestimated, as well as with an elastic-perfectly plastic or an elastic-brittle model, as shown in [12]. Indeed, as described in [12; 13], conventional approaches such as the linear Mohr-Coulomb (M-C) envelope or the non-linear Hoek and Brown (H&B) envelope, being shear-strength-based constitutive models, have limited ability to predict the depth/extent of brittle failure and to capture the behaviour of moderately jointed to massive rock masses where damage is the result of extensile fracturing. In these criteria, cohesive and frictional strengths are mobilised simultaneously. This is not the case in brittle failure, as frictional strength starts playing a role only when significant cohesion is lost [14]. Following this observation, [6] modified the conventional H&B criterion by proposing a brittle version with the H&B parameter \(m\) set to zero. Later, [PERSON]. [15] suggested a new constitutive model for brittle failure where the various strength components mobilise differently as a function of plastic strains (or damage), with a progressive cohesion loss due to tensile cracking and crack coalescence, followed by the mobilisation of frictional strength once the rock fragments can move relatively to each other in shear (CWFS, Cohesion Weakening Friction Strengthening). [PERSON]. [12] showed that this model satisfactorily predicts the failed zone depth and shape around the Mine-by test tunnel mentioned above. The CWFS approach was implemented in the widely used generalised Hoek and Brown criterion to numerically simulate brittle response in hard, massive rocks at great depths [16]. Thus, brittle failure can be well captured in continuum modelling by resorting to the so-called Damage Initiation Spalling Limit (DISL) approach. However, it was shown to be unable to distinguish areas with different damage intensities: the so-called Highly Damaged Zone (HDZ), where the material collapses, and the EDZ, where the material is fractured but does not collapse [13]. While revising empirical and numerical techniques to estimate spalling depth around underground openings, [PERSON]. [17] noted, recalling [18], that continuum modelling is clearly an approximation to the discrete fracturing process associated with spalling damage. Hence, another possibility is to adopt the Discrete Element Method (DEM) by [19] to capture combined intact material fracturing and discontinuity failure. Such a method treats the material as an assembly of separate blocks or particles that can move, rotate, and detach from one another until new contacts are established during the simulation. Thanks to the bonding between discrete elements, DEM can be used to simulate the fracturing of brittle materials. The advantage of DEM is that the complex empirical constitutive criterion is replaced with simple contact logic [18]. One of the significant areas for improvement in DEM is that fracture initiation and growth are generally not directly modelled but are implicitly accounted for using elastoplastic material constitutive laws assigned to the rock blocks [20]. New discontinuities cannot grow through continuum blocks, thus making it impossible to simulate the fracturing of intact rock and losing the pivotal role of micro-heterogeneities in the spalling failure of massive and brittle rocks.