Two numerical models were built. Both models describe a potential plane failure along a pre-existing joint. Model 1 presents a joint with an 80∘ dip angle, while model 2 presents a dip angle of 40∘. These two models have been proposed in agreement with the objective of this work: to study the phenomenology of the rock bridge failure. To do so, a steeply dipping rock wall and a gentle slope are considered. These two cases are defined in the function of the expected rockfall failure mode (shear or tension). It is expected that in the case of a steep slope, a tensile and/or shear failure mode will be observed. Indeed, authors (Stock et al., 2012; Bonilla–Sierra et al., 2015) have highlighted that the location of a rock bridge is important for understanding if rockfall fails in tension or shear, as they can form a pivot point about which the failing rock block is able to potentially rotate and fail in tension. In the case of a gentle slope, only a shear failure mode is expected. Therefore, it is possible to assess the influence of the location of the rock bridges as well as the initial morphology of the rock wall.

The geometry of the two models is presented in Fig. 2. The rock block presents a length of 6 m and a width of 1.5 m, leading to a total area of 9 m2, which, considering an out-of-plane thickness of 1 m, is also the volume (in m3) defined as “particularly dangerous for linear infrastructures and private residence” (Effendiantz et al., 2004). The total height of the model is 12 m. In practice, the geometry of the two models is the same; only the inclination of gravity is changed (angle α in Fig. 2).

During the meshing process, 128 contacts were created along the joint located between the block and the underlaying rock mass. Each contact can be defined by its coordinates in x and y (altitude). The behaviour of the rock joint is defined by the mechanical properties implemented for each individual contact (presented in Sect. 2.2). As only contacts belonging to regions can be modified in UDEC, the rock joint was then divided into 100 regions of the same length that can represent either “rock bridges” or “open-crack” areas. This division has been undertaken using the FISH language. Each region can therefore include one or two contacts. During the computation process, the local stress distribution along the joint can lead to the rupture of some rock bridge regions, then becoming a region of “failed rock bridges” that behaves as an open-crack area. This phenomenon progressively increases the number of open-crack regions along the joint.

Once the models are meshed, they are loaded only by gravity to evaluate the initial local state of stress along the joint.

An elastic model is assumed for the rock blocks, and a Mohr–Coulomb elasto-plastic model is assumed for the rock joint (contacts along the joint). A contact exhibits a shear failure mode when the local stress reaches the Mohr–Coulomb failure criterion and a tensile failure mode when its tensile normal stress becomes equal to the assigned tensile strength.

The mechanical properties of the rock blocks (Table 1) were defined based on a literature review of a common limestone in the French Alps (“Urgonian” limestone) (Frayssines, 2005). This limestone has been considered the reference in this study as it forms high cliffs in south-eastern France, where present traces of failed rock bridges are widely documented (Frayssines and Hantz, 2006).

Along the rock joint, three types of contact are considered:

rock bridge (RB) contacts which behave elastically with the same characteristics as the intact rock; to determine the normal and shear stiffness of the rock bridges, a centimetric opening of the joint has been considered;

The normal and shear stiffnesses of RB and RBF contacts have been defined based on a literature review of Urgonian limestone fractures (Frayssines, 2005). They are presented in Table 2.

The failure envelope properties of RB and RBF contacts (cohesion, friction angle and tensile strength) were defined following a step-by-step procedure. As the objective of the numerical modelling is to study the phenomenology of the rock bridge failure propagation, the failure criterion has to be close enough to the initial stresses along the joint, when considering only RB contacts. Therefore, during a first step, the distribution of stresses has been evaluated and compared to “classical” failure criteria provided in the literature (Frayssines, 2005). Then, in a second step, the characteristics of the criteria have been decreased to fit the objective. The classical values and the ones defined with this procedure for the RB and RBF contacts are presented in Table 3. Even if the values considered in the study are much lower than those found in literature, it is assumed that the failure propagation phenomenology will be the same as in reality. In the case of OC contacts all the values are taken as equal to 0 (Table 3) to ensure the phenomenology is identified and not polluted by other behaviour.

The modelling protocol proposed to study the rock bridge failure phenomenology is based on the following steps. It is summarized in Fig. 3.

All the 100 regions and so the 128 contacts of the rock joint are initially considered rock bridge (RB). In other words, 100 % of the rock joint is defined as RB. The model is run to equilibrium under gravitational loading. This corresponds to the initial stage (Step 0).

It can be noted that the numerical model has been validated by comparing the stresses evaluated by a simple theoretical analytical calculation of a block laying on an inclined plane by numerical shear and normal stress values.

To study the phenomenology of rock bridge failure (RB and RBF), the evolution of normal and shear stresses along the joint during the stepwise introduction of open-crack (OC) contacts has been analysed in detail. To do so, Scenario 1 was considered.

Figure 4 presents, for both model 1 and model 2, the distribution of the normal and shear stresses along the rock joint at different equilibrium steps 0 to n.

First, the distribution of the stresses along the rock joint is presented at Step 0, considering that the joint is only composed of rock bridges. In the case of model 1 (slope of 80∘), tension (σn<0) is observed at the upper part of the block (near point B in Fig. 4a). In model 2 (slope of 40∘), no tension is observed.

For both models, at Step 1, 10 % of the rock joint is intentionally modified from RB to OC contacts. In both models, the introduction of OC contacts results in a general increase in the shear stresses along the rock joint, with a stronger increase in these shear stresses in the vicinity of the OC area. This increase in the shear stresses brings the joint closer to the failure criterion not only in the vicinity of the OC area but also elsewhere, in particular at contacts located in the upper part of the rock joint (point B in Fig. 4). The normal stresses slightly vary during this first stage.

During each subsequent Step 2 to n, 2 % of additional contacts are modified from RB to OC in model 1 and 10 % of additional contacts are modified from RB to OC in model 2. These modifications induce the failure of rock bridges directly near the OC contacts by increasing the shear stresses along the rock joint, but the model reaches a mechanical equilibrium at the end of each step. There is also an increase in the normal stresses along the rock joint. This phenomenon continues until no mechanical equilibrium is reached anymore, which is associated with the downward sliding of the block (simultaneous failure of all the contacts).

The non-convergence of the model occurs when 16 % of the contacts are converted to OC in the case of model 1 and 30 % for model 2.

These results highlighted two phases during the rock bridge failure: a first phase during which only the intentionally created open-crack contacts are observed and a second phase during which the stress transfers induce the additional failure of rock bridges. In other words, in a first phase, the crack enlarges without inducing rupture elsewhere, and in a second phase the open crack reaches a state where rupture self-propagation starts until the block slides along the joint.

To study more specifically rock bridge cascading failure phenomenology, Scenario 2 was considered. Results are presented in Fig. 5 in terms of the proportion of so-called “failed contacts” (OC + RBF) versus the proportion of OC contacts along the joint. For both dip angles, there is a first linear phase during which the only failed contacts are the intentionally introduced OC ones. During this first phase, the block remains stable; i.e. a mechanical equilibrium is reached after each introduction of new OC contacts. Then, in a second phase, the redistribution of stresses caused by the introduction of new OC contacts induces the rupture of some RB contacts, which are converted into RBF contacts. During this second phase, even a small increase in the proportion of OC contacts leads to the rupture of additional rock bridges, which highlights the cascading failure phenomenology affecting the rock bridges. The slope of the linear regression in this second phase is around 10 in the case of model 1 and 5 in the case of model 2, meaning that the introduction of 1 OC contact leads to the failure of 10 RB contacts for model 1 and 5 RB contacts for model 2. This second phase starts for approximately 8 % of the rock joint defined as OC for model 1 and 23 % for model 2. The start of this phase differs slightly depending on the position of the RB and OC along the joint.

The non-convergence of the model starts when OC contacts represent 19 and 35 % of the joint for models 1 and 2 respectively.

Based on these preliminary results, Scenario 3 was considered. Results are shown in Fig. 6.

For both model 1 and model 2, two phases in the propagation of the rupture may be identified for all the simulations carried out. In the case of model 1, the second phase starts for an average of (10±2) % of the rock joint defined as OC, and the slide of the block (non-convergence of the simulation) occurs for an average proportion of (20±1.5) %. Regarding model 2, the second phase begins for an average of (29±5) % of the rock joint defined as OC, and the slide of the block occurs for an average proportion of (44±5) %. The transition area (Figs. 5–7) has first been identified in Fig. 6 and reported in Figs. 5–7. It corresponds to the transition between both phases in the propagation of the rock bridge failure and is due to the difference in the location of the RB.

In order to check whether there is a correlation between the two phases of rock bridge failure and the displacement that can be monitored on a potentially unstable block, a tracking point (C), shown in Fig. 7, has been introduced. Such a point could easily be instrumented in the real case of motion tracking if displacements of the order of millimetres are observed before the failure of the block.

Scenario 3 was considered. The displacement of point C was studied versus the proportion of OC along the joint, which is a marker of “virtual time”. The movement is no longer recorded as soon as all the contacts have failed, because the computation does not converge anymore.

Figure 7 shows that there is only one phase when considering the displacement. Both phases identified previously cannot be observed through displacement. To be certain of this result, a smaller mesh has been defined, and the same results have been obtained.

As highlighted by different authors (Tuckey and Stead, 2016; Stock et al., 2011), the location of the rock bridges has a strong impact on the stability of a potentially unstable block. To see whether our model leads to the same conclusion, the following protocol has been followed:

A number N of contacts is defined to be OC and randomly located along the joint. N is equal to 18 for model 1 (14 % of the joint) and to 46 for model 2 (36 % of the joint). These values were chosen for the model to be in the second phase, where the cascading failure phenomenology affecting the rock bridges is observed (Sect. 3.2). As seen previously, these proportions are sufficient to induce the additional RBF.

The results are presented in Fig. 8. The figure (top part) shows the values of the minimum, maximum and average contact altitude along the rock joint for both model 1 and model 2. It also shows (bottom part) the total number of considered failed contacts for a number N of contacts defined to be OC with respect to the average altitude of the OC contacts for both model 1 and model 2.

Figure 8a presents the results of model 1. It highlights that there are a larger number of failed contacts (OC + RBF) when the OC contacts are localized on average in the upper part of the joint. Figure 8b shows that, for model 2, contrary to model 1, there are a larger number of failed contacts (OC + RBF) when the OC are localized on average in the lower part of the joint.

This difference highlighted between the two models can be explained by the distribution of the stresses along the joint.

In model 1, there is tension in the upper part of the rock joint (Fig. 4a) when considering 100 % of RB contacts. To the contrary, in model 2, there is no tension along the rock joint (Fig. 4b).

In the presented study, as the tensile strength is relatively high in comparison with the cohesion and the friction angle, only shear failure was observed, and no tensile failure was reported. Based on this observation, it is needed to study more specifically the role of tensile strength in the evolution of RBF with time. To do so, a new model 3 has been defined and run. It is based on model 1 (dip angle of 80∘) as model 1 shows tension. In the new model, a tensile truncation was added to the Mohr–Coulomb failure criterion. The tensile strength (TS) has been taken as equal to the uniaxial compressive strength (UCS) value divided by 10 (UCS / 10). The compressive strength is calculated according to Eq. (1). 1UCS=2ccos⁡φ1-sin⁡φ, with c being the cohesion and ϕ the friction angle.

The mechanical characteristics of model 3 are listed in Table 4. The cohesion value has been increased in comparison to model 1 for numerical modelling requirements: when considering the same cohesion value, the model was not converging. The cohesion value has been increased until the model could be run.

The results are presented in Fig. 10. The tensile truncation of the Mohr–Coulomb failure criterion results in tensile failure of 6 % of the joint at the initial Step 0 (eight contacts present a tensile normal stress that becomes equal to the assigned tensile strength). At each subsequent step, 10 % of additional OC contacts are introduced along the rock joint. Because the cohesion is 3 times higher than for model 1, the stresses along the joint are further away from the failure criterion of rupture than for model 1 (Fig. 10). As observed previously, the normal and shear stresses progressively increase. It can be noted, as for the previous models, there is a more significant increase in the shear stress in the vicinity of the OC area. Up to 40 % of the joint can be defined as OC area before the calculation does not converge anymore.

For model 3, the transition phase identified previously is comprised of between 40 % and 50 % of the rock joint defined as OC, while in model 1, it is comprised of between 10 % and 20 %. In other words, when increasing cohesion value, the proportion of open-crack area needs to be higher to reach the cascading failure affecting the rock bridges than when considering low cohesion values. It justifies that in reality, as the cohesion values of the rock bridges are 500 times higher than in the study presented in this paper, only a few portions of rock bridges allow a potentially instable block to be in place. The second phase observed in the paper occurs instants before the fall of the block.

This study shows that, when considering tensile failure through the tensile truncation of the Mohr–Coulomb failure criterion, a proportion of failed rock bridges comes from the tensile stresses along the joint. However, the same “bi-phase” propagation failure phenomenology was observed regardless of the comprehensive consideration of the tensile failure.

In the modelling procedure presented in Sect. 2.3 and applied to models 1 to 3, the rock bridges that failed during the calculation (RBF) are considered to keep the same shear strength values as RB. This hypothesis has been made to consider asperity that can exist along areas of failed rock bridges. An alternate approach would be to consider that RBF contacts behave like OC contacts. This is discussed hereafter, by the mean of an additional model 4 comprising only two types of contact: RB and OC. RBF are considered to behave as OC. This model is based on model 1 (dip angle of 80∘), to which it will be compared.

The new OC contacts will be introduced into the upper part of the joint as it has been highlighted that for model 1, there are a larger number of failed contacts (OC + RBF) when the OC contacts are localized on average in the upper part of the joint.

Figure 11 presents the distribution of stresses along the joint at different steps of computation for models 1 and 4. The first OC area is introduced into the upper part of the joint, 10 cm away from point B. It is observed, as previously, that there is a general increase in shear stresses and a very small increase in normal stresses. Model 1 stops converging when 18 % of the joint is defined as OC (Fig. 11a), which is in agreement with what was observed before. When considering 16 % of the joint defined as OC (last step before the model does not converge), there are 22 % failed contacts (OC + RBF). Model 4 stops converging for 26 % of OC (Fig. 11b). Therefore, considering two or three types of contact gives similar results. To test this theory, Fig. 12 presents the proportion of so-called failed contacts (OC + RBF) versus the proportion of OC contacts along the joint. It shows that the two phases highlighted previously are again identified. The main difference comes from the fact that considering only two types of contact, the first phase is smaller.

In this study, the choice has been made to consider much lower strength properties of the rock bridges than in reality (see Sect. 2.2), due to numerical restrictions. Indeed, it has been shown by various authors (Frayssines and Hantz, 2009; Matasci et al., 2015; Tuckey and Stead, 2016) that only a few percent of rock bridges along the detachment surface are enough to maintain a compartment in a stable state. This extremely low proportion of rock bridges brings modelling challenges, such as high stress concentration at rock bridges, that have yet to be overcome. From a numerical point of view, modelling less than 1 % of the joint as rock bridges would require an extremely dense meshing, due to the high stress concentration and stress gradients in the rock bridge areas. In order to answer the objective of this paper, which is to highlight the phenomenology of the rock bridge failure propagation and not to accurately represent rock bridge behaviour, the authors have considered that decreasing the mechanical properties of the rock bridges is an adequate way of answering the presented difficulty.

Despite the fact that low rock bridge mechanical properties are imposed by numerical modelling restrictions, it is essential to assess the influence of this choice. The authors feel confident in the proposed methodology as models have been realized with higher mechanical properties and have shown similar phenomenology. An example of this is model 3, which considers in particular a higher cohesion value (130 kPa for model 3 compared to 45 kPa for model 1). Moreover, as shown by previous research, the compartment instability occurs through progressive fracturing of intact rock bridges, in a process termed step-path failure (Kemeny, 2005; Eberhardt et al., 2004; Scavia, 1995; Brideau et al., 2009) that may in some cases be compared to a cascade-effect failure: they can fail like dominoes along sloping channels (Bonilla–Sierra, et al., 2015; Harthong et al., 2012; Zhou et al., 2015). The study presented in this paper corroborates the previously observed cascade-effect failure of rock bridges.