Modeling Ellipsoidal Block Impacts by an Advanced Rheological Model

Mathematical expressions and unknowns vectors are formulated in the local reference system, in which normal and tangential directions (perpendicular and parallel to the slope, respectively) are considered (Fig. 1).

During the impact, block and soil interact with each other exchanging contact forces and contact momentum applied at the centre of gravity of the contact surface which evolves during the impact. This interaction between soil and block is modelled by means of an equivalent foundation, whose size is a function of block penetration and rotation, as well as the block shape. Therefore, the problem of evaluating the block motion during the impact and the force acting on the soil is split up in two sub-systems: (i) the rigid block motion and (ii) an equivalent foundation with its constitutive relationships.

The block motion is described by means of three degrees of freedom, the normal and tangential displacements of its centre of gravity and the block rotation around it assumed to be positive in clockwise direction. The block obeys the linear momentum balance equation written considering both the contact forces and the block self-weight. The linear momentum equation is the same as in Dattola et al. (2021) in which both the mass and inertia moment referred to the axis orthogonal to the motion plane \(n-t\) and passing through the center of gravity are updated considering the new block shape.

The constitutive macro-element relationships are based on the same rheological model proposed in Dattola et al. (2021) (see Fig. 2 for the normal impact case) in which the constitutive equations of each element are updated to take into account of the ellipsoidal shape without changing the model parameters apart the block dimensions. The updated equations are given briefly in the Appendices B and C.

The impact between the block and the soil is considered concluded, when at least one of the conditions between loss of adherence and toppling of the equivalent foundation is satisfied. In the former case, the block detaches from the ground surface and it flies, whereas toppling of the equivalent foundation takes place when the block touches the ground and the resultant of the contact forces lies on one side vertex of the equivalent foundation (i.e. the downward vertex of the equivalent foundation), which is also the point of instantaneous rotation, and the block starts rolling.

The linear moment equilibrium equation applied to the block, together with the compatibility and constitutive equations, forms a system of differential equations that admits a unique solution during block impact. The solution of this system provides the evolution of all contact variables, such as block displacement, velocity, acceleration, contact forces and energies.

Many authors (e.g., Bozzolo and Pamini 1986; Cancelli and Crosta 1994; Chau et al. 2002) demonstrated that the block geometry affects the impact process and the restitution coefficients, by controlling both bouncing velocity and block angular velocity. In the proposed rheological constitutive equations, the geometry is taken into account synthetically by means of three functions of the block displacement component: the tangential and out-of-plane dimensions of the equivalent foundation \({B}_{t}={B}_{t}({u}_{n}^{bl},{\theta }^{bl})\) and \({B}_{w}= {B}_{w}\left({u}_{n}^{bl},{\theta }^{bl}\right)\), the normal and tangential arms of the contact force components, \({r}_{n}={r}_{n}({u}_{n}^{bl},{\theta }^{bl})\) and \({r}_{t}={r}_{t}({u}_{n}^{bl},{\theta }^{bl})\), respectively (see Fig. 17 in the D.4 Appendix D) where \({u}_{n}^{bl}\) and \({\theta }^{bl}\) are the normal displacement of the block centre of gravity and the block rotation, respectively.

In appendix \(\mathrm{C}\), the above functions are derived for ellipsoidal blocks (Fig. 3), starting from those for spherical (di Prisco and Vecchiotti 2006), prismatic and cylindrical blocks (Dattola et al. 2021). For the sake of simplicity, the cross-section orthogonal to the major axis of ellipsoid is assumed to be circular with a radius of \({R}^{bl}\) (Fig. 3). The block shape is described by the aspect ratio \({A}_{r}={H}^{bl}/{R}^{bl}\) (i.e. \({A}_{r}=1\) represents a spherical block).

In principle, the model can take into consideration the out-of-plane dimension, related to the block mass value and to the out-of-plane pseudo-foundation dimension, affecting the block motion and the restitution coefficients. Nevertheless, for the sake of simplicity, this factor is not discussed in the following, since as was already clarified in Fig. 3, the authors considered exclusively rotational ellipsoids (i.e. with a circular section).

The block initial orientation (\({\theta }_{0}^{bl}\)) is the angle between the major semi-axis and the normal direction.

In this section, a mathematical relationships connecting the apparent normal and tangential restitution with the energetic restitution coefficient is derived. The obtained expression highlights also the role played by the impacting angle, the rotational kinetic energy ratio before and after the impact.

Let us consider an impact with the following assumptions: (i) the block impacts on a planar substrate with a fixed inclination; (ii) no dissipation is due to the air drag (i.e. air resistance is considered to be negligible if compared with the dissipated energy during the impact) and (iii) differently from Bozzolo and Pamini (1986), in which the instantaneous rotation centre coincides with the contact point, we does not impose the position of instantaneous rotation centre, (iv) the block is rigid so that no block shape and/or size changes are admitted and, finally, (v) neither fragmentation, nor spalling and chipping are considered, implying the block mass and inertia moment do not change during impact. During impact the block total kinetic energy reduces due to dissipation. If it is assumed that the z-coordinate value of the block centre of gravity does not change just before and after the impact (i.e. the block potential energy does not change), then the variation in the block mechanical energy is due only to the change in kinetic energy. Since block rotation is enabled the kinetic energy is the sum of the translational kinetic energy \({E}_{\mathrm{kt}}\left(t\right)\) and the rotational kinetic energy \({E}_{\mathrm{kr}}\left(t\right)\). Furthermore, during impact the block centre of mass moves in a vertical plane, then it is possible to split up the velocity vector in the normal (\({v}_{n}\)) and tangential (\({v}_{{t}})\) components and, in turn, by using the apparent normal (\({e}_{n}=-{v}_{n}^{f}/{v}_{n}^{0})\) and tangential (\({e}_{{t}}\)) apparent restitution coefficients, the velocity components can be written in terms of \({v}_{n0}\) and \({v}_{t0}\), the normal and tangential components (before the impact), respectively. The translational kinetic energy can be written as:where \({E}_{\mathrm{kt}0}\) is the initial value of the translational kinetic energy, and \({\theta }^{in}\) is the impact angle, (i.e. the angle between the initial velocity vector and the ground normal) (Fig. 3b). The rotational kinetic energy ratios at the beginning and the end of impact are:andin which \({E}_{\mathrm{kr}0}\) and \({E}_{k0}\) are the initial rotational kinetic energy and the initial kinetic energy, respectively. The kinetic energy is written as:and, therefore, the following formulation for the energetic restitution coefficient (Heidenreich 2004) is obtained:

When both block shape and size are taken into account, the above definition of restitution coefficients is not sufficient. In fact, if the restitution coefficients are referred to the velocity components at the centre of mass, they embed both the ground material dissipation and the block rotation effects. For this reason, Vijayakumar et al. (2012), in their analysis based on the rigid body approach, distinguished the effective normal from the apparent normal restitution coefficient. The first one was defined by using the normal velocity components at contact point, whereas the second one at the gravity centre of the block. In the case of an elliptical block, under the assumption of instantaneous impact, rigid block and rigid impacted material, these authors found that the apparent restitution coefficient \({e}_{n}\) is given by the following expression:being \({e}_{n}^{*}\) the contact restitution coefficient. In this formula the geometrical effects are included in the inertia moment \({I}_{\mathrm{bl}}\), the initial block configuration in the horizontal arm \({r}_{t}\) and the block angular velocity in the \({\dot{\theta }}_{0}/{v}_{\mathrm{bl}0,n}\) ratio.

Vijayakumar et al. (2012) proved, by using the above expression, that:

As previously mentioned, block geometry and orientation at impact play an important role, in affecting both block penetration and generated force, the main variables in the design of rockfall protection structures. In the parametric analysis, spherical blocks were also considered for comparison since this shape is commonly employed in rockfall analyses.

In Fig. 4a, b, the evolution with time of vertical displacement and vertical force are reported, considering blocks with different aspect ratios impacting along the major axis. The vertical displacement trends are characterized by an increment up to a peak. Then, blocks with a low value of the aspect ratio stop while blocks with higher aspect ratio experience a vertical displacement reduction till block detachment from the soil. Both peak vertical displacement and time increase with \({A}_{r}\).

The maximum force reduces with increasing \({A}_{r}\), while the peak time increases with \({A}_{r}\).

A comparison between blocks impacting vertically with major and minor axes with different \({A}_{r}\) values was performed and the results were plotted in terms of maximum vertical displacement in Fig. 5a and maximum vertical force in Fig. 5b.

For all the aspect ratios, the blocks impacting along the major axis exhibit larger values of the maximum vertical displacement, that increases with \({A}_{r}\), and the same trend is observed for impacts along the minor axis. The maximum vertical displacement of the spherical block is the minimum of the two trends. The maximum vertical forces are affected by the block \({A}_{r}\) for both the orientations (Fig. 5b) and in particular, they decrease with \({A}_{r}\) for blocks impacting with both the minor axis and the major axis.

For inclined impacts, the problem is not symmetric anymore and block rotation takes place coupling the block displacements, since normal and tangential forces generate momentum. Under this condition, the impact process depends on block shape, orientation at impact (\({\theta }_{0}\)) and impact angle (\({\theta }^{in}\)). The impact angle (\({\theta }^{in}=20^\circ\)) and the initial block orientation (\({\theta }_{0}=0^\circ\)-impacting along the major axis) were fixed and results are reported in Fig. 6, where block rotation and rotational energy ratio (\({R}_{r}\)), defined as the ratio between rotational (\({E}_{\mathrm{kr}}\)) and total kinetic energy (\({E}_{k}\)), were used. The trends of vertical displacement and force, as a function of \({A}_{r}\), coincide with those relative to vertical impacts (Fig. 6a, b).

During impact, the rotation increases monotonically (Fig. 6c). The rotational energy ratio increases monotonically until a maximum. Subsequently, blocks with lower shape ratios experience a slight reduction.

In Fig. 7, the results obtained by considering different impact angles, \({\theta }^{in}\), and \({A}_{r}\) are illustrated. The maximum vertical displacement reduces with increasing the impact angle and increases with \({A}_{r}\). On the contrary, the maximum vertical force reduces with increasing \({A}_{r}\) and for increasing impact angles. For increasing \({A}_{r}\) values, the increment of block rotation reduces apart from the spherical block, whose value is intermediate. Furthermore, the trend of block rotation with \({A}_{r}\) is characterized by a peak for all \({A}_{r}\) values. The evolution of the maximum rotational energy ratio with impact angle is characterized by a peak and its value increases with \({A}_{r}\), although this increment is less evident for higher values of \({A}_{r}\).

The proposed model describes in detail the impact process, but in practical problems, global variables such as apparent normal and tangential restitution coefficients, maximum penetration depth and block rotation are usually employed or required for defensive work design. In particular, restitution coefficients make the simulations for extended stochastic modelling more efficient.

It is well known (Asteriou et al. 2012; Asteriou and Tsiambaos 2018; Asteriou 2019) impacts and relative coefficients of restitution are influenced, among the others, by block geometry, size and orientation, impact angle and velocity. In this section, the effects of block geometry and orientation, as well as \({\theta }^{in}\) on the global variables were investigated.

In Fig. 8, the effects of block orientation and \({A}_{r}\) on global variables are shown. Normal restitution coefficient for blocks impacting with the major axis (Fig. 8a) increases with \({A}_{r}\), whereas an opposite trend is observed for blocks impacting along the minor axis. Furthermore, the normal restitution coefficients for blocks impacting with the major axis are larger than the normal restitution coefficient for blocks impacting along the minor axis while the spherical block takes an intermediate value.

When \({\theta }_{0}^{bl}\) (Fig. 8b) is considered, a symmetric trend is observed with respect to the 90° and a “swallow wings” trend is observed and the difference between maximum and minimum values of normal restitution coefficient, obtained by changing the block orientation, increases with \({A}_{r}\). Analogous consideration holds true for the maximum penetration depth (Fig. 8c).

When the line between contact point and the center of gravity is not aligned to either major or minor axes, even a vertical impact produces, as shown in Fig. 8d, a variation in block rotation. The variation in the rotation is antisymmetric because if the initial block orientation causes the center of mass to move to the right with respect to the normal then the block rotates clockwise. In the opposite condition, the block undergoes an anti-clockwise rotation. The variation of the block orientation, for the same \({\theta }_{0}^{bl}\), increases with \({A}_{r}\) until 2.5 then it reduces. This is due to two antagonistic factors: the increase in \({A}_{r}\) causes not only an increase in \({r}_{t}\) (tangential arm) but also an increase in block penetration. The former effect facilitates the block rotation, the latter one prevents it.

In Fig. 8e, the evolution of the maximum normal force with the initial block orientation and for different \({A}_{r}\) values is reported. The trend is symmetric with respect to 90° (i.e. impact along the minor axis) and for this limit value the influence of the A_r is reversed (i.e. larger A_r corresponds to lower maximum normal force). The maximum absolute value of the contact moment versus the initial block orientation, for different aspect ratios, is shown in Fig. 8f. The trend is symmetric with respect to the 90° initial block orientation for lower values of \({A}_{r}\) a clear trend is visible but for higher values of \({A}_{r}\) no specific trend is observed.

When inclined impacts of blocks aligned along the major or the minor axis are considered, the results, in terms of normal and tangential restitution coefficients, shown in Fig. 9 are obtained. When \({A}_{r}\) increases normal restitution coefficients decrease whereas an opposite trend is observed for the tangential one (Fig. 9a, b). For \({A}_{r}>1,\) the trend of normal restitution coefficient is not monotonic and for \(0<{\theta }^{in}\le 50^\circ\) becomes negative. \({e}_{n}\) is these cases negative since \({e}_{n}\) is calculated as a function of the center of mass velocity, whereas the detachment between block and soil is a function of the contact velocity. For sufficiently large of \({\theta }^{in}\) values the normal restitution coefficient rapidly increases since the plastic slider activates at the very beginning of the impact. In contrast, when the blocks impact with the minor axis, both coefficients decreases with increasing \({A}_{r}\) (Fig. 9c, d).

Numerical simulations were also performed by assuming \({c}_{\mathrm{v}}=0.5\), \({\gamma }_{\mathrm{v}}=1.70\) and \({\gamma }_{\mathrm{v}}=0.85\). The results are shown in the supplementary materials. The observed trends are qualitatively similar to those discussed in this section. These parametric numerical simulations were performed to investigate the effects of viscous parameters on kinematic and dynamic variables.

As already stated, rockfall simulations describe blocks trajectories during impacts, free fly, rolling and sliding motions, ending up when the block kinetic energy is very small. The assessment of the block trajectory allows to compute important variables for the design of protection structures, such as the maximum trajectory height, the run-out distance and the maximum block velocity.

These variables are also important rockfall hazard mapping, and they are controlled by slope inclination, block geometry and orientation, angular velocity, normal and tangential restitution coefficients. To study the influence of block geometry and initial angular velocity on the trajectory, the HyStone numerical simulation code (Agliardi and Crosta 2003) was extended by implementing the proposed impact model. In order to focus the attention on block geometry and its angular velocity, a bi-planar topography was used (Fig. 10), with a \(230\) m slope height (\({H}_{{s}}\)), inclined at \(\omega = 30^\circ\) with respect to the horizontal plane (horizontal projected distance \({L}_{s1}=400\) m) and a final \(600\) m long horizontal plateau (\({L}_{s2}\)). The initial block translational velocity was \(20\) m/s inclined of \(10^\circ\) with respect to the horizontal axis, while, in agreement with Caviezel et al. (2021) and Buzzi et al. (2012), the initial angular velocity ranged from \(0^\circ\)/s to \({20{,}000}^\circ\)/s. To study the block geometry effects, \({A}_{r}\) was varied between \(1.0\) (sphere) and \(4.0\) (markedly elongated ellipsoid), by keeping constant the block mass m = \(100\) kg, whereas both initial block translational and rotational kinetic energies were the same for all the numerical simulations.

The impact parameters were maintained as those of the previous simulation (reported in Table 1).

The first investigated parameter was \({A}_{r}\), by assuming blocks to be initially oriented along the major axis and initial angular velocity to be nil. The results, together with the slope profile, are reported in Fig. 11. Since initial conditions (i.e., velocity, angular velocity and orientation) coincide and motion equations during the first free-flight are independent of block shape, all blocks (Fig. 11a) trajectories superimpose for \(x<250\) m. Except for the spherical block, after the first impact, blocks start rolling. The spherical block starts rolling after three small bouncing (Fig. 11b, c). All the blocks reach the horizontal plane where they stop at different distances (Fig. 11c).

As is well known, the maximum vertical elevation of falling blocks is essential to design position and dimensions of protection structures (e.g. fences, nets and embankments). For this reason, in Fig. 11b, block elevation histories are illustrated. Again, only spherical block elevation history differ from the others.

The translational velocity moduli, referred to the centre of mass, histories are reported in Fig. 11c: four phases are evident:

Apart from the spherical block, the results relative to the different blocks markedly differ when the horizontal plane is reached (along the inclined plane the translational velocity differs but the scale employed in the plot does not allow to capture such a difference).

The evolution of the rotational velocity component is illustrated in Fig. 11d. The rotational velocity components are constant when blocks fly, whereas at impact, they vary abruptly, since contact forces generate moments. Finally, during the rolling phase, the rotational velocity components increase linearly when the blocks move along the inclined slope and decrease with parabolic trends until the blocks stop, when they move along the horizontal plane.

Subsequently, the role of initial angular velocity has been investigated for all the previous \({A}_{r}\) values (in this case all blocks are oriented along the major axis). The initial angular velocity was varied in the range 0°/s–20,000°/s, a range sufficiently wide in the light of the real case data available in the literature (e.g. Buzzi et al. 2012; Caviezel et al. 2021).

The values of the maximum vertical elevation above the slope profile for each portion of ballistic trajectory and for all the block trajectories are expressed in terms of number of blocks in Fig. 12.

As is evident, the trend of these histograms and data scatter markedly depend on block shape, even if, in general, the number of blocks reduces progressively with vertical elevation.

In the supplementary material, the results obtained by performing the same simulations but by using \({c}_{\mathrm{v}}=0.5\) and \({\gamma }_{\mathrm{v}}=1.70\) and \({\gamma }_{\mathrm{v}}=0.85\) are reported.

Block shape role in affecting impacts and bouncing (Bozzolo and Pamini 1986; Azzoni et al. 1995; Vijayakumar et al. 2012; Leine et al. 2014; Glover 2015) is largely recognized in the scientific literature. As stated by Yan et al. (2018), the effects of both block shape and orientation are usually neglected, due to the difficulties to study irregular shapes, lithology and rock mass subdivision (Fityus et al. 2013). Block shape affects impact force and penetration depth, normal and tangential restitution coefficients and, thus, block velocity after the impact. These, in turns, are fundamental to simulate block trajectory and, consequently, to locate and design correctly protection structures and to prepare hazard maps, when using models based on restitution coefficients. As well, the contact force and its maximum value at impact are fundamental in the design of protection structures in case of direct impact on the structure or on a cushion layer interposed between block and structure. The effects of the geometry on the contact force was investigated by Yan et al. (2018), by means of the finite element method (LS-DYNA code), in which the contact option is enabled to study the interaction between a concrete slab and the ellipsoidal block. Block shape was described through the sphericity index \({S}_{{p}}\), defined as:making possible a comparison with our results. Yan et al. (2018) computed, the evolution of the impact force for three block orientations and different impacting velocities, but for the same block mass. They found that a reduction in \({S}_{{p}}\) (i.e. passing from spherical to ellipsoidal blocks) or, equivalently, an increase in \({A}_{r}\), induces a reduction in the maximum impact force. We confirmed this result for all impact inclinations and \({A}_{r}\) values, although the impacted material is different (Figs. 6, 7b). Shen et al. (2019, 2020) performed DEM simulations with ellipsoidal and irregularly shaped blocks constituted by a clump of particles, impacting on horizontal layers made up of the same particles used for the block. They confirmed the decrease in the maximum impact force for increasing \({A}_{r}\) values and for different falling heights (i.e. velocity), in agreement with our results (Fig. 5b).

As far as block penetration is concerned, Shen et al. (2019) demonstrated its increase with \({A}_{r}\). This was confirmed by our simulations (Fig. 6a) for vertical impacts with vertex (Dattola et al. 2021), for inclined (Fig. 7a) and for vertical impacts (Fig. 8c) and different orientations of ellipsoidal blocks. Hence, our results allow a more general description of impact and could definitely support the design of countermeasures.

As mentioned above, modelling based on coefficients of restitution is still adopted in many analyses and modelling codes. Experimental and in situ results showed that normal restitution coefficients can be larger than one, especially when the angle between block velocity vector and ground surface normal (i.e. the complementary of the impact angle) is very low. This effect was attributed to block rotation, increasing the normal restitution velocity component in comparison with the corresponding one at impact. Since the amount of rotation is controlled by the arms of the normal contact force components, elongated blocks markedly show this effect. Generally, the proposed solutions started from the hypothesis of a point-like contact around which the block rotates (Bozzolo and Pamini 1986; Azzoni and De Freitas 1995; Azzoni et al. 1995; Glover 2015). Usually, restitution coefficients can be computed by employing different approaches being the rigid body impact mechanics the most popular. According to it, both body and contact plane are rigid, and the deformations, developing in regions near the contact point, are taken into account by a dummy element, virtually interposed between block and contact plane (Ashayer 2007). The equations have been expressed in impulse terms with additional assumptions needed to solve the problem (see Bozzolo and Pamini 1986; Descoeudres and Zimmermann 1987; Azzoni et al. 1995). Unlike our approach, no constitutive equations describing the impacted material were proposed. This implies irreversible deformations were not considered and, therefore, block sinking into the contact plane (material) is not calculated. In addition, the absence of constitutive equations prevents the calculation of the contact forces. Hence, in our approach, the penetration of the block into the ground surface introduces a major improvement in the model capabilities strongly influencing the results.

To highlight the relationship among block rotation, contact geometry and apparent restitution coefficients, the contact restitution coefficients are compared with apparent restitution coefficient values calculated by using our impact model (Eq. 6) and the Vijayakumar et al. (2012) model (Fig. 13). The comparison is accomplished for different initial block orientations and initial rotational velocities, in the case of an ellipsoidal block with \({A}_{r}=2.0\). In case of no initial rotation, all the graphs (Fig. 13a) are symmetric with respect to the initial block rotation at \(90^\circ\). This symmetry is lost when blocks have an initial rotational velocity and the degree of asymmetry increases with its intensity (Fig. 13b–d). The apparent restitution coefficients, both for the proposed and the Vijayakumar et al. (2012) model, become negative (i.e. the normal velocity after the impact referred to the block centre of mass is directed downward, nevertheless the normal velocity at the contact point is directly upward) for a large range of initial block orientations and initial angular velocity. On the contrary, both contact and energetic restitution coefficients are always positive. In fact, the contact coefficient is computed considering the normal velocity at contact point which is always positive since the blocks bounce. This is not true for the apparent restitution coefficient even if the blocks bounce. In fact, the block rotational velocity at the exit is so high that causes a reversal of the normal velocity at the centre of mass where the apparent coefficients are computed. Therefore the contact velocity after the impact is directed upward while the normal velocity at the centre of mass, due to the rotation, is directed downward. In case of no initial rotational velocity, all the coefficients are the same for blocks impacting with the major or minor axes, because in these cases, the blocks do not generate rotational velocities. In case of spherical blocks, apparent and contact restitution coefficients coincide since the block rotation does not affect the normal velocity at the centre of mass.

Rigid body contact mechanics was used by Ashayer (2007) in case of ellipsoidal blocks to obtain normal restitution velocities, normal restitution coefficient and restitution angular velocity. Under the same initial conditions, the results agree both in terms of normal restitution coefficient/normal velocity and rotation with those reported in Fig. 8.

Ashayer (2007) performed also DEM numerical simulations of ellipsoidal blocks constituted by a clump of circular particles by means of a modified particle–particle and particle–wall contact law. By increasing the number of clump particles the results converge to those obtained by performing rigid body simulations and, thereby, to our results. This furtherly supports the capabilities of the here proposed model.

Bozzolo and Pamini (1986) proposed a rigid body model for ellipsoidal blocks calculating translational and rotational velocity components after impact by imposing rigid rotation at the contact point and conservation of angular momentum. They validated the model by discussing the experimental results of three real rock falls. Although geometry and parameters are different from those employed by the authors, their number of blocks distribution of the maximum ballistic height decreases similarly to our results (Fig. 12) and the same holds for simulations by Azzoni et al. (1995).

Finally, with the reference the numerical results obtained by employed HyStone code and describe in Figs. 11 and 12, the authors assessed \({e}_{n}\) and \({e}_{t}\) for all impacts. The number of blocks distributions of all these values are plotted Fig. 14. The higher value of number of blocks have normal restitution coefficient in the range from 0.75 to 1.0 (mean values of \(\mu =0.67\)) and a tangential coefficient from 0.7 to 1.1 (\(\mu =0.92)\). As already observed by Bourrier et al. (2012) for in situ tests, owing to block rotation and block elongated geometry, both coefficients the number of blocks follows a normal distribution and both coefficients can reach values well above 1.