# Application of Bonded-Block Models to Rock Failure Analysis
## 1 Introduction  
The safety assessment of underground structures in rock relies on the capability to analyze the response of the rock mass to the processes of excavation and support installation, as well as the effect of subsequent actions throughout its lifetime. Numerical modeling tools have steadily progressed in the representation of the non-elastic phenomena that govern the deformation and failure of rock masses. While equivalent-continuum models remain an important tool in many practical engineering situations, discontinuum models that explicitly incorporate the rock mass joints, possibly including the development of new fractures, are an attractive tool for the simulation of failure processes. The numerical techniques classified as Discrete Element Methods (DEMs) comprise a wide range of formulations and computer codes, where the underlying concept is the representation of geomaterials as systems of particles, grains or blocks. [PERSON] [1] proposed numerical models of rigid polygonal or circular particles in which the mechanical response was governed by the constitutive relations of the contacts between discrete bodies. DEM models became an important tool in rock mechanics, allowing the study of complex block systems defined by rock mass discontinuities, joints, bedding planes and faults. The inclusion of block deformability through the use of internal finite element meshes became common in many problems. It allowed the non-elastic behavior or the intact rock material to be considered. An alternative approach involved the application of the discontinuum concept at a finer scale in order to analyze the internal behavior of the rock blocks [2]. The study of rock fracture by means of \"Bonded-particle models\", composed of assemblies of circular or spherical particles, became a key investigation topic, allowing the simulation of complex responses of rock by means of relatively simple constitutive relations at the contact level [3]. The inclusion of rock mass discontinuities produced the \"Synthetic rock mass\" concept [4]. Advances in computational power have enabled the use of more complex shapes for the component particles, namely, polygonal or polyhedral, leading to the \"Bonded-block model\" (BBM) framework [5; 6]. The progressive development of these models, aiming at more closely simulating physical fracture processes, has led to the generation of larger and more elaborate block assemblies, as well as more complex contactrelations. Reviews of the multiple numerical formulations, considering both particle and block models and discussing their advantages and drawbacks, are available [7; 8; 9; 10]. The present paper addresses the application of bonded-block models to underground works in rock; discusses the essential assumptions, the model generation and analysis procedures, and the results that can be obtained; and provides guidelines for their application. The proposed approach is based on a fairly simple numerical model using rigid blocks and a standard DEM contact formulation. The contact fracture model employs a bilinear post-peak softening curve that can be calibrated by given fracture energies. It is shown that this type of model is capable of reproducing the modes of fracture observed around underground openings, namely, spalling patterns. The context of the problem is introduced in the next section by means of a literature review of the key developments in the discrete element modeling of rock fracture. Section 3 discusses the fundamental issues involved in the use of bonded-block models. The generation of block shapes using Voronoi or Delauney algorithms is addressed, and the outcomes of the two alternatives are compared. The procedure for the calibration of the micro-properties for a given randomly generated assembly using the experimental data is described. The role of contact constitutive models including fracture mechanics concepts is discussed.