An Analytical Model to Predict the Survival Probability of Irregular Brittle Rocks in Rockfall under collinear impact

The experimental survival probability of one irregular shaped rock was established via 105 drop tests using mortar replicas. The derivation and validation of an analytical model to predict the survival probability of brittle rocks of irregular shape upon collinear impact is presented. The survival probability predicted by the model was found to fall withing 5% of the experimental data with an excellent goodness of fit coefficient (R2 ~ 93%). The experimental survival probability of one irregular shaped rock was established via 105 drop tests using mortar replicas. The derivation and validation of an analytical model to predict the survival probability of brittle rocks of irregular shape upon collinear impact is presented. The survival probability predicted by the model was found to fall withing 5% of the experimental data with an excellent goodness of fit coefficient (R2 ~ 93%).


Introduction
Predicting possible fragmentation during a rockfall event is necessary to optimise the design of protective structures and to adequately mitigate the rockfall risk. For a given geological setting, estimating the likelihood of fragmentation is essential. This can be quantified via the survival probability. Recently, the authors produced the first model that can predict the survival probability of brittle spheres in free fall against a horizontal and planar surface, given the statistical distribution of material properties (Guccione et al. 2021a). The model was validated on mortar spheres of different diameters and using two mortars. The authors subsequently identified which parameters have the most influence on the prediction of survival probability (Guccione et al. 2021b) and proposed a methodology to remove outliers from the Brazilian test results that constitute the model inputs (Guccione et al. 2022). Although this model is a breakthrough, its applicability to rockfall is still limited by the fact that it assumes spherical rocks. This technical note presents the derivation and validation of an improved version of the model for brittle rocks of irregular shapes, making the model more relevant to rockfall engineering.

Rationale
The model for irregular rocks presented herein is based on the model developed by the authors for brittle spheres (Guccione et al. 2021a) where the key model inputs are derived from a series of Brazilian tests performed on discs (of mortar or rocks) and analysed in terms of survival probability of work at failure and of force at failure. These two statistical distributions capture the variability of material properties and are used to derive four key input parameters, namely the critical work W cr BT , the shape parameter of the Weibull distribution of work at failure m BT−W , the shape parameter of the Weibull distribution of force at failure m BT−F and the critical force at failureF cr BT . The critical Brazilian work W cr BT is then converted into a critical kinetic energy E cr k via three conversion factors ( C shape ,C size ,C rate ). The variability of force required to reach failure, expressed via m BT−F , appears in C size and the critical force at failure F cr BT appears in C shape .

3
In the model published by the authors in 2021, the only source of variability is that of material properties, because for spheres impacting a planar surface, the position of the sphere is irrelevant and the impact conditions are repeatable. When an irregular rock impacts a surface, the position of the rock at impact will affect the likelihood of fragmentation. In the revised model, this is captured by considering a range of possible impact positions and assuming that the object will fail in indirect tension with a single vertical (or sub-vertical) crack emanating from the point of impact (see Fig. 1). Such assumption is based on experimental observations for rocks failing under a collinear impact, made during this research and previous tests by the authors (Guccione et al. 2021a). For an irregular rock, the fracture area created at impact hence depends on the position of the rock at impact, which introduces an additional degree of variability that can be captured by considering a distribution of possible cross sections given the rock shape. This geometrical variability is then combined to the material variability, as presented in the next section.

Model Derivations
Let us consider a disc (diameter d , thickness d∕2 ) and an irregular block having a volume equal to that of a sphere having a diameter d . Both objects (block and sphere) are made of the same material and have the same strength. Assuming a linear evolution of force with displacement during the Brazilian test, the work required to fail a disc can be approximated as: where F d is the force at failure during Brazilian test conducted on a disc and d is the corresponding displacement. Similarly, the work required to fail a brittle object of irregular shape in indirect tension ( W is ) is written as: where the force and displacement at failure are now noted F is and is . Consequently, we get: It is not possible to analytically compute the deformation of an irregular object in compression and the solution derived by Japaridze (2015) for a disc is cumbersome for the present derivation. So, for a matter of simplicity and elegance, let us assume that both the disc and the irregular rock deform under load according to Hertz's formulation for spheres (Stronge 2000): For an irregular rock in compression, the deformation is assumed to be governed by the height of the cross section in the direction of loading H is : (4) and (5) gives: For a Brazilian test on a disc, the force at failure and tensile strength are related via: where A d is the cross-sectional area of the specimen tested in Brazilian test ( A d = d 2 ∕2 ) and F d is the force at failure. For an object of irregular shape, in absence of analytical solution, the equation for a sphere is assumed to hold: where A is is the cross-sectional area of the irregular rock being failed in compression and F is is the force at failure. We then get: Fig. 1 Fragmentation of a mortar sphere (a) and irregular rock replicas (b and c) upon impact on a concrete slab The term 5∕3 W BT d −3 pertains to the Brazilian tests on the discs while the term A is pertains to the geometry of the irregular rock. As discussed in Sect. 2.1, W BT , A is and H is are not unique values, there exists a range of values for a given material and a given rock. Given the statistical distribution of W BT and of A is 5∕3 H is −1∕3 , it is possible to obtain the distribution of W is . This is achieved by considering all possible combina- . Once all these combinations have been calculated, the cumulative distribution of work at failure of the irregular block ( CD(W is ) ) and the corresponding survival probability ( SP(W is ) ) can be obtained, using a package like Excel. It is relevant to recall that, for any data The survival probability ( SP(W is ) ) is then fitted with a Weibull distribution and the two Weibull parameters, W cr is and m Wis , are identified. These two parameters are key inputs of the model, as shown in Eq. (16) and in Fig. 2.
The conversion factor for size and rate, C size and C rate , are then applied to W cr is to obtain the critical kinetic energy of an irregular rock of similar shape but larger volume: The equations for C size and C rate are those established by Guccione et al. (2021a) and are recalled here, for clarity of presentation: where d is the diameter of the disc tested in Brazilian test, D is the diameter of a sphere having the same volume that the irregular rock, and m BT−F is the Weibull distribution shape parameter for the distribution of force at failure for Brazilian tests on discs.
where is is equal to 0.8; Ỹ x−y is an equivalent modulus at the contact between material x and material y with subscript r , s and surf referring to rock, steel and (impacted) surface, respectively. Ỹ x−y is defined as: with Y x representing the Young's modulus of material x and x its Poisson's ratio. ISR is the increase of strain rate defined as: where t BT is the time to reach failure in a quasi-static Brazilian test and t impact is the duration of impact (including the compression phase and restitution phase) for a drop test.
The central part of the survival probability, which drops from 100 to 0% with kinetic energy of the irregular object of equivalent diameter D ( E K(D) ), is derived as per Guccione et al. (2021a) and is expressed as: (16) Guccione et al. (2021b) conducted a sensitivity analysis and showed that the values of Poisson's ratio and ISR only have a marginal effect on the prediction of survival probability. In contrast, the moduli and the distribution of Brazilian test results strongly influence the prediction. Figure 3 shows the distributions of work (a) and force (b) at failure for Brazilian tests, of geometrical data ( A is 5∕3 H is −1∕3 ) (c) and of work at failure of the irregular rock W is (d). This later distribution is obtained by combining the work W BT and geometrical data A is 5∕3 × H is −1∕3 , as per Equation [10]. Details about the rock used for this study is given in Sect. 3.
The distribution of W is can be turned into a survival probability and fitted with a Weibull distribution to determine its shape and scale parameters. Values of 3.58 and 3.947 were found for m Wis (shape) and W cr is (scale), respectively.
All input parameters for the model are given in Table 1. ) for the irregular rock; d distribution of work required to fail the irregular rock in Brazilian test (W is )

Preparation of Rock Specimens and Distribution of Cross Sections
An equant rounded rock of about 5 cm in size was selected for this study. The rock was scanned using a EinScan Pro 2 × Plus scanner to obtain a 3D mesh. Using the software Meshlab (Cignoni et al. 2008), a homothetic change of dimensions was applied to the rock to obtain a volume equivalent to that of a 50 mm diameter sphere, for comparison purposes with the data from Guccione et al. (2021a). The dimensions of the scaled rock along the three axes are given in Table 2. From the scanned rock, 3D printed plastic moulds were created as per Guccione et al. (2021c) and about 100 mortar replicas were made. The mortar composition is that used by Guccione et al. (2021a): it consists of a mixture of sand, cement, lime and water in 3:1:0.25:1 ratios. The replicas were cured in water for 8 weeks and left to dry for another 4 weeks. Such mortar composition and curing process results in an average unconfined compressive strength of 22.9 MPa and an average secant elastic modulus of 4.2 GPa.
To establish the distribution of cross section areas ( A is ) and section heights ( H is ), a Python script using the library trimesh (Dawson-Haggerty et al. 2019) was written. The idea is to rotate the rock mesh around its three principal axes at 10° increments and, for each position, the point of impact with a horizontal planar surface was identified (point of lowest elevation). From that point, a series of vertical cross sections in 10° increments were created (black lines in Fig. 4b). For each cross section, H is was taken as the maximum height of the section (see Fig. 4c).

Fragmentation Cell for Drop Tests
The fragmentation cell developed by Guccione et al. (2021c) was used for all drop tests conducted for this study. For conciseness, the device is not described in here and the reader is referred to Guccione et al. (2021c) for more information. The irregular rock specimen was held above the instrumented concrete slab, at a predefined and measured height, by a flexible pipe in which vacuum was applied. Four high speed cameras were used to record the impact on the concrete slab and the occurrence of fragmentation. The photographs are used to distinguish cases where the rock breaks at the first impact from situations where it takes a second impact (after the first rebound) to break the rock. In the later cases, the rocks were considered as unfragmented at first impact. Different rock orientations were used when positioning the rock specimens in the holding system (vacuum pipe) to achieve some variability of impact position. Indeed, to compare the experimental results to the model, it is important to achieve some variability of the impact position, given that the model considers a wide range of cross sections.  Table 2 Geometrical characteristics of the rock used for this study.
Roundness and sphericity were calculated as per Cox (1927) and Wadell (1933), respectively  Guccione et al. (2021a) tested mortar spheres of 50 mm in diameter and found that the survival probability dropped from 100 to 0% as impact velocity increased from about 6.5 m/s to about 8.5 m/s. Based on this information and the fact that the central part of the survival probability is approximately linear, it was decided to conduct three drop tests for each rock, at impact velocities of 6, 7 and 7.5 m/s, corresponding to kinetic energy of 2.3, 3.1 and 3.5 J, respectively. For each impact velocity, about 35 rocks were dropped.

Results
Given the input parameters of Table 1 and using Eqs. (11), (12) and (13), the size conversion factor, rate conversion factor and critical kinetic energy for the survival probability in drop test can be computed. Values are reported in Table 3. Using Eq. (16), the central part of the survival probability takes the form: Figure 5 presents the experimental and predicted (Eq. (17)) survival probability for the irregular rock tested in this study. As expected, the experimental survival probability function of the irregular rock is flatter than that of the sphere of same volume, which reflects more variability. Indeed, the variability of material strength is now combined to the geometrical variability.
In terms of model performance, the predicted survival probability falls very close to the experimental data points, both in shape and scale. For all three experimental values of kinetic energy, the absolute difference between the model and the data is in the order of 5% and the goodness of fit indicator R 2 is equal to 92.6%.
The predicted critical kinetic energy for the irregular rock is 3.3 J against 3.52 J for the sphere of same volume and mortar (Guccione et al. 2021a), which is only a 6% difference. Similarly, the difference in slope of the central part of the predicted survival probability between the irregular rock and sphere is only in the order of 10%. Such small differences come from the fact that the irregular rock tested is equant with a sphericity of 0.92. On the other hand, the experimental survival probability of the sphere is clearly distinct from that of the irregular rock and the model is good enough to capture such difference. Note that using a less equant object will mostly likely result in an eccentric impact, which the model does not account for, at this stage.

Conclusions
Rock fragmentation upon impact during rockfall is one of the most challenging phenomena to understand and predict. To date, there is no fully predictive model in the literature that can predict the likelihood or outcome of fragmentation of natural rocks. The authors recently proposed and validated an analytical model that can predict the survival probability upon impact of brittle spheres. The model is based on statistical information gathered from a series of Brazilian and unconfined compression tests. Although the model represents a breakthrough in fragmentation prediction, it only applies to spheres and is therefore of limited use. This paper presents an extension of the previous model that captures shape variability and can predict the survival probability of a brittle irregular rocks in drop tests under collinear impact.
The model combines variability of material strength and variability of impact position by considering a range of cross-sectional fracture areas. It is assumed that the irregular object fails in indirect tension upon a single collinear impact. At this stage, the model does not account for double impact or eccentric impact. A series of experimental drop tests on mortar rock replica were conducted to offer a preliminary validation of the model. The predicted survival  probability falls very close to the experimental data, which demonstrates an excellent predictive capability of the model. Note that the fact that mortar is used to test the model, for control and repeatability reasons, is not an inherent limitation of the model. Technically, the model can be applied to any brittle material.