# Application of Bonded-Block Models to Rock Failure Analysis
## 3 Generation and Application of Bonded-Block Models  
### Modeling Assumptions and Block Patterns  
A bonded-block model is composed of an assembly of discrete blocks, defined by two types of discontinuities: (i) the real rock mass discontinuities, namely, joints, faults and other features; (ii) a random network of potential discontinuities inside each intact rock block corresponding to conceivable crack paths. Figure 1 shows the conceptual representation of three rectangular blocks divided into Voronoi-shaped polygonal blocks, hereafter named \"inner-blocks\" (or sub-blocks). The joints between the rectangular blocks stand for real rock mass discontinuities. The interfaces between the inner blocks are initially bonded and are assigned the strength of the rock matrix. They can progressively fail in tension or shear, allowing the progressive fracture and fragmentation of the block. Voronoi patterns resemble the grain structures of many types of rock, and thus, various authors have made inner blocks that correspond to different minerals. In this case, it is possible to refine the model by subdividing each Voronoi polygon into triangles built around a central point, thus allowing different properties for radial and peripheral joints to distinguish between inter-grain and intra-grain cracks [35]. Alternative shapes of inner blocks have been investigated. It is also possible to create an inner-block structure by means of a Delaunay algorithm, a triangular mesh built by connecting the centers of adjacent Voronoi polygons. In most instances, the inner-block pat  
Figure 1: Bonded-block model. Three rock blocks discretized with a pattern of bonded polygonal inner blocks. as a means to create a network of potential cracks with random orientations. Cracking patterns in Voronoi-shaped models often resemble those observed in tensile failure. In 3D, tetrahedral inner blocks can also be built with a Delaunay triangulation algorithm [6]. These triangular or tetrahedral patterns produce a crack network with more continuity than the Voronoi shapes, since several planes meet at the nodes, generally allowing mechanisms of failure involving more extensive shearing and sliding. The interaction between inner blocks typically follows the standard DEM representation based on point contacts. For matching edges, two point contacts are located at the endpoints. In the case of Figure 1, along the horizontal and vertical joints, a number of vertex-edge contacts are created to analyze the mechanical interaction between the blocks. The inner blocks may be numerically represented by either rigid or deformable blocks. In the former case, the total deformation takes place along the joints, which are assigned appropriate stiffness parameters. The option for deformable blocks may be implemented by creating an internal mesh of elements (sometimes designated as zones). The simplest solution relies on uniform-strain finite elements, either triangles or tetrahedra, which can be easily fit into arbitrary block shapes. Fine internal meshes require extra nodes to be placed along the edges, which will also contribute to improving the stress distribution along the joints between the blocks, as they provide more contact points. The mechanics of block contacts are described in the following section. The bonded-block concept is intended to represent the fracture of the rock matrix by means of tensile cracking and shearing along the interfaces between the inner blocks. Failure under compressive loads is also accounted for by the progressive fracturing of the inner-block interface network. In the analysis of a real rock mass, the pre-existing discontinuities also have to be included. Major features can be inserted at known locations, and joint sets can be defined by a few representative discontinuities. Discrete fracture networks may also be used based on statistical parameters obtained in field studies.