Measuring and Predicting the Angle of Repose of
Granular Matter from Clump and Potential Particles
DEM Approaches

V. Angelidakis, S. Duverger, S. Nadimi, S. Utili, S. Bonelli, P. Philippe,

Jérôme Duriez

► To cite this version:

V. Angelidakis, S. Duverger, S. Nadimi, S. Utili, S. Bonelli, et al. Measuring and Predicting the Angle of Repose of Granular Matter from Clump and Potential Particles DEM Approaches. TC 105 International Symposium 2024, TC105 Geo-Mechanics from Micro to Macro, Sep 2024, Grenoble, France. pp.012090, 10.1088/1755-1315/1480/1/012090. hal-05031402

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MEASURING AND PREDICTING THE ANGLE OF REPOSE OF GRANULAR MATTER
FROM CLUMP AND POTENTIAL PARTICLES DEM APPROACHES

V. Angelidakis<fnref n="1" />,<fnref n="2" />✉, S. Duverger<fnref n="3" />, S. Nadimi<fnref n="2" />, S. Utili<fnref n="2" />, S. Bonelli<fnref n="3" />, P. Philippe<fnref n="3" /> and J. Duriez<fnref n="3" />

**Abstract:** The angle of repose (AoR) is a widely employed measurement for granular materials that is currently used by various industries to characterise the behaviour of powders, soils and grains. This study proposes a new measurement technique for the AoR, and applies it on numerical models of two recently developed devices to measure the AoR by means of Discrete Element Simulations, for plane-strain and axisymmetric repose states. 3D-printed rounded tetrahedral particles are simulated using two numerical modelling strategies, namely concave multi-spheres (clumps) and convex potential particles, to explore the effect of particle shape on the AoR, and comparisons are drawn against experimental evidence. Then, a campaign of drained triaxial compression tests is carried out for a wide range of confining pressures to investigate if any correlations exist between the AoR and constitutive (peak and critical-state) friction angles.

**1. Introduction**

Granular materials exhibit different phases depending on their loading and boundary conditions, as they can flow like a complex fluid or stabilise in a heap configuration like a solid. Their stable re-pose state is typically characterised by a linear slope for the sake of simplicity, i.e. a constant angle of repose (AoR), even though such a description can bear simplifications with respect to reality. The ability of numerical Discrete Element Methods (DEM) to predict the AoR has recently been benchmarked in a round robin series of tests [1, 2] organised by the Japanese Geotechnical Society (JGS), within the activities of Technical Committee 105 (TC105: Geo-Mechanics from Micro to Macro) of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE). Following our recent work [3], DEM results obtained with the YADE code [4] are herein presented, together with new measurement methods which can inform future AoR studies.

**2. Numerical simulations of AoR tests**

The experiments of the benchmark study organised by the JGS applied to mono-sized 3D-printed rounded tetrahedral particles with given visco-elastic and frictional properties [1]. While the particle shapes at hand can be modelled accurately using multi-sphere particles, i.e., rigid aggregates of overlapping spheres, simulations have also been conducted using convex three-dimensional potential particles [5], as shown in Figure 1, to quantify the importance of particle convexity and sphericity on the AoR. In addition to particle shape, the set-up of the DEM simulations conformed the experimental configurations in terms of overall geometry (i.e. “plane-strain” and “axisymmetric”), as well as statistical distributions of interactions properties (i.e. interparticle friction, and coefficient of restitution) and total number of particles. The initial positions of the particles were randomised in the simulations, which were ran repeatedly and interpreted in a statistical fashion, to enhance the robustness of the conclusions.

**3. Measurement of AoR in numerical simulations**

Measuring the AoR (denoted as $\alpha$) is a subjective process, affected both by user dependency and by the complexity of the repose state geometry. In the round robin study organised by the JGS

<figure><img src="image_2.png" /></figure>

<footnote n="1">¹ Queen's University Belfast, School of Natural and Built Environment, Belfast, United Kingdom</footnote>

<footnote n="2">² Newcastle University, School of Engineering, Newcastle upon Tyne, United Kingdom</footnote>

<footnote n="3">³ Aix Marseille Univ, INRAE, RECOVER, Aix-en-Provence, France</footnote>

<footnote>✉ v.angelidakis@qub.ac.uk</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 1. (a) Plane-strain and (b) axisymmetric repose states, considering models with concave multi-spheres (clumps), and models with convex potential particles.</figcaption></figure>

[1], the AoR was calculated using the coordinates of particles at the extremes of the repose state, i.e. the elevation z of the apex particle at the top of the repose state and the horizontal distance L of the farthest particle at its base, as the angle $\alpha = \arctan(z/L)$. For the axisymmetric repose state, various L measurements were taken to find the minimum, average and maximum AoR [1]. We will refer to this method as a “local” AoR measurement, as it is affected by the local conditions at the extreme particle locations.

To improve on this approach, which only considers a few extreme particle coordinates, we propose a calculation of the AoR as the gradient of a regression curve fitted to the coordinates of the surface particles. First, the particles at the surface of the repose state are identified and their coordinates are projected on a vertical plane to form a 2D profile of the repose state, a process that differs for the plane-strain and the axisymmetric repose states, since the latter requires the consideration of polar coordinates [3]. As shown in Figure 2, the repose state can be approximated by a 1st degree (linear) polynomial regression curve ($\tilde{y}_1 = a_1\tilde{x} + b_1$), in which case the AoR is calculated as $\alpha = \arctan(a_1)$ or by a 2nd degree (quadratic) polynomial regression curve ($\tilde{y}_2 = a_2\tilde{x}^2 + b_2\tilde{x} + c_2$), in which case the AoR is calculated as the local angle $\alpha(\tilde{x}) = \arctan(2a_2\tilde{x} + b_2)$ that varies with distance $\tilde{x}$ from the origin of measurement. To achieve meaningful AoR measurements, some particles are excluded from this calculation, such as particles affected by boundary effects near the base of the repose state or particles near its apex (see [3]), where the repose state may flatten, as shown in Figure 2. We will refer to this proposed method as a “global” AoR measurement, as it takes into account all the particle coordinates forming the surface of the repose state, and not just the ones near its extreme coordinates.

Table 1 compares the local and global AoR measurements for DEM simulations of each AoR device, using concave (clump) particles for different initial packing conditions. It is found that the proposed global method leads to higher AoR values than the local method [1] for the plane-strain repose tests, and lower values for the axisymmetric repose tests. Thus, a systematic trend was not found between the two measurement techniques, while they both demonstrate similar degrees of standard deviation.
<figure><img src="image_4.png" /><figcaption>Figure 2. Angle of repose $\alpha$ using 1st and 2nd degree polynomial regression for the (a) plane-strain and (b) axisymmetric repose states of concave (clump) particles [3].</figcaption></figure>

<table><caption>Table 1. Comparison of AoR measurement methods on the set of clump-based simulations. Numeric values correspond to average, standard deviation, and minimum and maximum values.</caption><thead><tr><th></th><th>Plane-strain</th><th>Axisymmetric</th></tr></thead><tbody><tr><td>Global AoR measurement</td><td>36.6° ± 1.0° ∈ [34.1°; 37.8°]</td><td>37.0° ± 0.8° ∈ [35.4°; 38.5°]</td></tr><tr><td>Local AoR measurement</td><td>38.1° ± 1.1° ∈ [35.0°; 41.3°]</td><td>33.9° ± 0.8° ∈ [32.0°; 36.1°]</td></tr></tbody></table>

## 4. AoR for concave clumps and convex potential particles

Despite their visible morphological differences, the concave clumps feature similar values of convexity and sphericity with the convex potential particles. The clump particles have a convexity of 0.954 and a degree of true sphericity of 0.933, whereas the potential particles have a convexity of 1 and a degree of true sphericity of 0.985. Convexity is defined as the ratio of the particle volume over the volume of its convex hull, and the degree of true sphericity is defined as the ratio of the surface area of a sphere with equal volume to that of the particle, over the surface area of the particle itself.

A comparison between the AoR from DEM simulations and experiments yields interesting results, where the local measurement method of the AoR was used, as it was the only one possible during the experiments. First, Table 2 shows that the clump-based simulations predict the AoR within an 8% tolerance, even after calibration of the particle-scale parameters to match the experimental ones. This could be attributed to various reasons, such as the consideration of an idealised contact model in the DEM simulations, the real surface roughness of the 3D-printed particles that was not simulated, or the complexity of simulating friction in general. Second, it appears that neglecting the local concavities of the physical particles using the potential particles leads to further underestimation of the AoR, which is to be expected, as convex particles cannot mechanically interlock compared to the concave clumps. Hence, even for these fairly convex rounded tetrahedra, using purely convex potential particles led to an underestimation of the AoR by 4°, compared to the clump results.

## 5. Comparison of AoR with macroscopic friction

This study finds no relation between AoR values and void ratio for neither of the repose states, while the AoR value is found to be protocol-dependent, i.e. it varies for the plane-strain and axisymmetric repose states. Two novel techniques were developed to measure the void ratio at the repose states, introduced in [3]. This indicates that although the AoR is used for material characterisation in practice, it does not really present a constitutive nature, i.e. the AoR is a macroscopic
<table><caption>Table 2. AoR values from experiments and DEM simulations using the “local” measurement method.</caption><thead><tr><th></th><th>Plane-strain</th><th>Axisymmetric</th></tr></thead><tbody><tr><td>Experiments</td><td>41.4° ± 1.3° ∈ [38.3°; 46.3°]</td><td>35.3° ± 0.9° ∈ [33.3°; 37.3°]</td></tr><tr><td>Clumps</td><td>38.1° ± 1.1° ∈ [35.0°; 41.3°]</td><td>33.9° ± 0.8° ∈ [32.0°; 36.1°]</td></tr><tr><td>Potential Particles</td><td>34.8° ± 1.6° ∈ [32.5°; 38.0°]</td><td>29.7° ± 0.8° ∈ [28.5°; 31.2°]</td></tr></tbody></table>

system observation, rather than a constitutive material property. To further explore this, we conduct a series of drained triaxial tests for the clump particles, considering a wide range of confinement pressures, to compare the AoR against the macroscopic peak and critical-state friction angles $\phi = \arctan((\sigma_1 - \sigma_3) / (2\sqrt{\sigma_3 \cdot \sigma_1}))$, where $\sigma_1$, $\sigma_3$ the major and minor principal stresses of the triaxial tests. No correlation is found between them, as shown in Figure 3(a).

Lastly, Figure 3(b) compares the void-pressure relation of the repose states against the critical state line (CSL) obtained via the triaxial tests, with the latter calculated both for the whole system and after removing the rattlers, i.e particles with less than two contacts, as they are not part of the load-bearing mechanism of the system and may bias the CSL [6]. It is shown that the void ratio for both plane-strain and axisymmetric repose states does not show any particular trend with the mean pressure, while it appears to be bounded by the CSL with rattlers and the CSL without the rattlers, an observation that requires further investigation.

<figure><img src="image_5.png" /><figcaption>Figure 3. (a) AoR $\alpha$ vs $\phi$ and (b) void ratio $e$ vs mean pressure $p$ for fully-formed repose states compared to the critical state line (CSL) for confining pressures $P_c$ with and without rattlers [3].</figcaption></figure>

## References

[1] Nakata, Y., et al. (2022). Experimental data of 3D printed granular material for verification of discrete element modeling simulation. *Soils and Foundations*, **62**, 101178.

[2] Saomoto, H., et al. (2023). Round robin test on angle of repose: DEM simulation results collected from 16 groups around the world. *Soils and Foundations*, **63**, 101272.

[3] Duverger, S., Angelidakis, V., Nadimi, S., Utili, S., Bonelli, S., Philippe, P., and Duriez, J. (2024). Investigation techniques and physical aspects of the angle of repose of granular matter. *Granular Matter*, **26**, 20.

[4] Smilauer, V., et al. (2021). *Yade Documentation 3rd ed.* The Yade Project, http://yade-dem.org/.

[5] Boon, C. W., Houlsby, G., and Utili, S. (2013). A new contact detection algorithm for three-dimensional non-spherical particles. *Powder technology*, **248**, 94–102.

[6] Duverger, S., Duriez, J., Philippe, P., and Bonelli, S. (2021). Rattlers’ involvement for possibly looser critical states under higher mean stress. *EPJ Web of Conferences*, **249**, 11002.