# Application of Bonded-Block Models to Rock Failure Analysis
## 2 Discrete Element Modeling of Rock Fracture
### Bonded-Block Models  
Discrete element block models have been mainly employed in rock engineering to study failure modes defined by rock discontinuities. Researchers have taken advantage of the versatility of the method, creating randomly generated joint systems or detailed block assemblies, in order to examine stress distributions in rock masses [18], dam foundation problems [19; 20] or local effects on slope stability [21; 22]. Subsequently, it was recognized that the development of new fractures could be addressed by defining potential crack paths, initially bonded, which could then fail under the applied loading. This type of simulation of cracking and fracture was used, for example, to study underground waste storage projects [23; 24] and to investigate fundamental issues of crack propagation [25]. The suitability of randomly generated Voronoi polygons/polyhedra to represent the internal structure of rock materials has been recognized in various studies [26]. These geometries resemble the observed patterns in some way, so the block assembly may be used to reproduce the grain shapes and size distributions of the various mineral components [5; 27; 28; 29; 30]. These block models are substantially more demanding in computational terms than circular-particle models, mainly due to the effort to detect and analyze the contact between interacting complex shapes. Therefore, 2D models of laboratory specimens were the first applications, but 3D models and field problems are presently feasible. The designation of \"bonded-block models\" (BBMs) for these block assemblies, with the intention to study the fracture of geomaterials, was adopted more recently [6; 31]. The blocks are defined by a random network of potential cracks, forming triangles/tetrahedra or Voronoi polygons/polyhedra, which are initially bonded but allow the fracture process to progressively develop [9]. Three-dimensional Voronoi patterns were used to analyze uniaxial compression tests on anisotropic rocks [32], as well as triaxial tests on rock salt, relating the damage evolution characterized by the breakage of bonds with experimentally measured acoustic emissions [33]. Different geometries of the joint network have been investigated, for example, a triangular-block geometry in 2D models, allowing it to break into three parts [34]. Another way to generate a triangular-block geometry it to split the Voronoi polygons into triangles formed by connecting each vertex to a central point [35]. Therefore, the initial grains are able to break along these potential internal cracks. A 3D model of tetrahedral blocks can also be obtained by means of a Delauney algorithm, which was employed, for example, in the analysis of the fragmentation evolution in veined rock around underground openings [6]. Other numerical formulations, namely, those based on FDEM or FEM/DEM approaches [36; 37], have been used in rock fracture studies. The underlying concepts are similar. The main differences with respect to DEM codes are typically related to the implementation of contact mechanics, their numerical treatment and the constitutive assumptions. For example, FDEM models were used in the analysis of crack development around underground excavations in rock [37]. A different BBM formulation in which fictitious stress is estimated inside the rigid blocks, which are allowed to split along an arbitrary plane, was applied in a study of the fracture of transversely isotropic rocks [38]. Alternative numerical implementations of fracture mechanics formulations have been proposed for analyzing failure processes in underground works [39] or slopes [40]. A reference should also be made to research on more complex block shapes, namely, involving smoothed boundaries [41].