# Application of Bonded-Block Models to Rock Failure Analysis
## 3 Generation and Application of Bonded-Block Models
### Constitutive Model with Post-Peak Softening for Inner-Block Bonds  
The mechanics of the contacts along the fictitious joints can be approached with various constitutive models. The simplest assumption is to assume brittle failure, governed by the peak values of tensile strength, cohesion and the friction angle. These micro-properties can be calibrated to obtain the intended global strength. More elaborate models add a post-peak-softening regime in tension and shear, which allow the consideration of fracture energies [37; 38; 55; 56]. Research on fracture modeling has shown the importance of size effects due to the numerical element size. In practice, the contact discretization in bonded-block models cannot be made fine enough to eliminate these effects, but the inclusion of fracture energies in the bond breakage criteria may reduce them. Experimental data are available in the literature to characterize the fracture mechanics parameters for different types of rocks [59; 60; 61].  
In the present BBM, the contact model proposed by [PERSON] and [PERSON] [56] is applied. In this formulation, the post-peak regime can be represented by a piecewise linear curve defined by a set of points. These authors compared linear, bilinear and trilinear softening curves, concluding that the intermediate case provides a sufficiently accurate approximation in practical applications. Therefore, in the examples presented, a simple bilinear softening curve is adopted in tension and shear, as shown in Figure 2. In the normal direction, the vertical axis represents the joint normal stress ratio, defined as the ratio of the tensile stress to the peak tensile strength. The horizontal axis is the joint normal displacement ratio, defined as the joint normal displacement over the normal displacement at peak stress. The first post-peak point corresponds to a fraction (\(\beta_{\text{n1}}\)) of the peak tensile stress, and the second point corresponds to the residual stress, here assumed to be null. The corresponding displacements are defined by the parameters \(\alpha_{\text{n1}}\) and \(\alpha_{\text{n2}}\), multipliers of the peak elastic normal displacement. In compression, elastic behavior is assumed, using the same joint normal stiffness. In the shear direction, a similar type of post-peak curve is adopted, with the vertical axis denoting the ratio of the shear stress to the peak cohesive strength. The post-peak softening of the cohesive strength is defined by the parameters \(\beta_{\text{s1}}\), \(\alpha_{\text{s1}}\) and \(\alpha_{\text{s2}}\), as shown in Figure 2b. The friction angle is assumed to remain unchanged; thus, after the cohesive strength reaches the null value, the contact reverts to Coulomb friction. The areas under the curves measure the fracture energies involved in the failure processes in tension and shear.  
The tensile strength parameters can be calibrated by performing a numerical uniaxial tensile test. Typically, in tension, the micro- and macro-properties are not very different; thus, the experimental strength value is often taken for the fictitious joints. The numerical uniaxial compression test is the critical one to calibrate the bond strength in shear. In Section 4.1, these numerical tests are analyzed for the examples presented.  
It should be noted that BBM models typically represent compressive failure by simulating the progressive cracking and slip of the bonds along the random joint network. In compression, the contact constitutive model behaves elastically. Some authors have developed contact constitutive models that include failure in compression, for example, using cap-type formulations [62]. These models typically require the assumption of small displacements, as the overlap of the blocks in contact may create numerical difficulties.