# Application of Bonded-Block Models to Rock Failure Analysis
## 3 Generation and Application of Bonded-Block Models
### Mechanical Micro- and Macro-Properties  
As in bonded-particle models, a distinction is made between \"micro-properties\" and \"macro-properties\" [3]. Macro-properties are the properties of the global rock mass, defined by its deformation and strength parameters. Micro-properties are the material parameters required by the numerical model to control the mechanical behavior of inner blocks and their interfaces, the potential crack network. The size chosen for the inner blocks influences the global mechanical response of the system, as the joint density changes. Therefore, for a given block size, the micro-properties have to be calibrated to produce a global response consistent with the intended macro-properties, which are provided by laboratory experiments on rock specimens or field tests on the rock mass. The choice of the inner-block size depends on the computational resources available and the desired refinement of the analysis, considering, for example, the size of the underground excavation and the expected extent of failure zones. Once the block size is selected, calibration is performed using numerical specimens generated with similar block dimensions. The deformability of the numerical model depends on the elastic moduli of the inner blocks and the stiffness of the joints or interfaces between them. The global deformation of the random-shaped block system should match the experimental data. There are different ways to split the deformation between the block material and the fictitious discontinuities. The simplest option is to assume the inner blocks to be rigid so that the entire block deformation is produced by the inner joints. At the other end, it is possible to adopt the desired Young's modulus of the block for the inner blocks. In this case, the joints will have to be given a very high stiffness so that they do not influence the deformation. In explicit time-stepping solution algorithms, unreasonably high stiffness parameters penalize computational efficiency. Intermediate solutions are possible, for example, by assigning 9095% of the deformability to the blocks, which gives approximately the same results without excessive run times [56]. In most bonded-block models, the inner blocks are assumed to be either rigid or elastic. It is certainly possible to assign non-elastic behavior to them, for example, by means of elasto-plastic or damage constitutive relations. However, this choice increases the complexity of the analysis and the calibration procedure. The dominant approach is to enrich the numerical representation response by refining the potential crack patterns, keeping the block or particle behavior as simple as possible. The calibration of micro-properties typically starts with the definition of the deformability parameters in the elastic range. The target Young's modulus can be that of the rock specimen, for example, in the simulation of lab tests, or the rock mass deformation modulus for field applications. The numerical joints in DEM models are zero-thickness interfaces, with the constitutive relations defined in terms of joint stresses in the normal or shear direction and the differences in the corresponding displacements across the joint. In the elastic range, the ratios of stress to relative displacement define the joint normal and shear stiffness. For the case of rigid inner blocks, a first approximation of the normal stiffness of the fictitious joints is given by \(\mathrm{kn}=\mathrm{E/d}\), where \(\mathrm{E}\) is the target Young's modulus, and \(\mathrm{d}\) is the average size of the inner blocks. Similarly, the shear stiffness may be estimated from the target shear modulus \(\mathrm{G}\) as \(\mathrm{ks}=\mathrm{G/d}\). These expressions are only a first estimate. A numerical test on a sample of the block arrangement with the assumed block size provides a better approximation. Typically, a few random systems are tested, and the average value is adopted.