数理解析研究所講究録 1413 巻 2005 年 75-83

75
Dependence of the angle of repose of heaps on the particle shape
Salah A. M. El Shourbagy, Hans-Georg Matuttis, University of ElectrO-Communications,
Department of Mechanical Engineering and Intelligent Systems, Chofu, Chofugaoka 1-5-1,
Tokyo 182-8585, Japan
Abstract
We investigate the dependence of the angle of repose for heaps of dry granularmaterials for heaps built from polygons which flow out of ahopper. We report the parameter dependence on inter-particle friction and particles shape.
For any elongation, the angle of repose depends crucially on the coefficient
ffiction. For non-elongated particles, the angle of reposeofdepends crucially on the number of corners of the particles.
$(\mathrm{d}\mathrm{r}\mathrm{y}/\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{o}\mathrm{m}\mathrm{b})$1Introduction
In this article we investigated the different parameters influence on theangle of repose. In both experiments
and simulation via the discrete ele
ment method (DEM), the angle of repose, afundamental quantity for theexperimental classification of granular materials, depends crucially on
the particles properties. Previoussimulations for the angle of repose of
dry granular materials used compositions of round particles and neglected friction $[1, 2]$(see Fig.1) or the rotational degree of freedom [3]. As aresult, the slopes of the heaps,
were not smooth and straight, but rough and curved, which is typical for
not for dry granular materials [5, 6, 7]. These simulations resorted to using rough grounds
to retain acertain verisimiltude, though heaps of dry sand can be built even on mirrors,
which are “microscopically” flat. The models neglect friction an rely on the interlockingof grain surfaces, so that rough grounds are necessary to obtain plausible looking results.
$\mathrm{w}\mathrm{e}\mathrm{t}/\mathrm{c}\mathrm{o}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{v}\mathrm{e}$2Setup of the Simulation
We simulate the sand heap in two dimensions as an aggregate of polygons. The particles have three degrees of freedom, two for the linear motion, one for the rotation. Thepolygons are inscribed into ellpses with varying length, for non-elongatffi particles intocircles. The particles are simulated as “soft” particles, so that the force between neighboring particles is computed proportional to their overlap area and their Young modulus.
Further components of the inter-particle forces are the coefficient of restitution $<1$ andthe Coulomb ffiction coefficient. The forces are used for the Integration of Newton’s
Equations of Motion, the integration by Gear Prffl.ctor-Corraetor/Backward Difference
Formula (for details, see Ref. [4])76Figure 3: Initial state (left) and end configuration (right) of a simulation for the computation of the outflow from a hopper on a non-rough ground with friction coefficient$\mu=0.6$ . The slopes are straight, as in experiments with non spherical particles. Particles
in the same layer have the same shade of gray, mixing of shades in the end configuration
indicates mixing of layers.

2.2

Simulation Geometry

The particle are initialized above a hopper geometry (see Fig.3) and fall down at thebeginning of the simulation. During the simulation, they aggregate on the hopper surface
and slide down the slopes of the hopper. The walls which limit the simulation geometry
and which form the hopper have for simplicity the same simulation parameters as theparticles, i.e. the same Young modulus, coefficient of restitution and friction coefficient

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Figure 4: Center of mass for particles on left and right slopes $(+)$ and least squares fit(o) for the determination of the angle of repose.

2.3

Computation of the angle of repose

Because the slopes in our simulations are straight, the angle of repose can be obtainedby a least-squares fit (see Fig. 4) to the center of maae of the highest particles in eachparticle diameters length. The left and right slope of each realized heap
were(initial particle configurations computed from different random number sequences, i.e.
the shapes of the particles, but not the size distributions, were slightly different for eachrun) were treated as independent configurations.
$\mathrm{o}\mathrm{f}\approx 0.5$

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Figure 6: Final configuration of a simulation of particles with five comers, and smoothground with ffiction coefficient $\mu=0.6$ .
coefficients, also sMing on the ground
can occur, for large coefficients of friction
the buildup of the heaps takes only place
via avalanches on the heap surface.
For non-elongated polygons with fivecomers, we computffi the dependence of
the angle of repose on the ffiction coefficient (see Fig. 7). The average angle ofrepose increased linearly between $\mu=0.2$
and saturated beyond $\mu\geq 0.5$ .
The reason for the saturation is, that theparticle rotation and ffiction “compete” ascauses for the motion of the particles: For
vanishing ffiction coefficients, the angle ofrepose can be extrapolatffi to 0 within er- Figure 7: Dependence of the angle of repose
ror bars, consistent with the truism that on the coefficient of friction for nearly-regularmaterials with vanishing Coulomb ffiction pentagons.
(fluids) don’t fom heaps.
$\mathrm{a}\mathrm{n}\mathrm{d}\mu=0.4$

4Particle roughness

Though the literature of granular matter abounds with simulations of round particles, inthis section we want to investigate the effect of deviation from the round shape in theabsence of a particle elongation, which, in a certain way, is the roughness of the particlae.

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Figure 9: Final configuration of a simulation with non-elongated polygons with 25 comers
md coefficient of ffiction $\mu=0.6$ .

80.

$\iota$’’’$$l,\backslash t^{\mathit{4}}$$..t.\backslash -*$$k<$$l_{P}^{\mathrm{e}}d.\ovalbox{\tt\small REJECT}\tau$$\dot{J}$$r_{\yen^{\tau}}^{\subset}.r^{r}$$‘ i<,’\sqrt$$;_{\gamma,\backslash },.\backslash _{\mathrm{s}}$.

$h_{\wedge,\nearrow},$$\#.‘\ovalbox{\tt\small REJECT}_{\mathrm{Y}}$.

$_{\mathrm{k}}\grave{\mathrm{t}}t\delta\chi’\swarrow$$\mathrm{t}$$arrow d$

“

,$.m$

$\sigma j\mathrm{r}-nF^{k_{\mathrm{c}}}\swarrow’,\swarrow$,’$\dot{\cup}\mathrm{Q}B-\mathrm{k}f^{V_{*}}\dot{\iota\epsilon}_{\mathrm{k}}$$\overline{\prime}\swarrow\gamma\cdot$$\mathrm{k}^{\gamma}\prime n_{u/}\prime^{J^{\mathrm{v}^{P’}}}arrow-$$\mathrm{k}.w,r^{J’}\dot{r}\mathrm{r}^{x^{\prime^{<_{\mathrm{r}}}}}\infty.\backslash$$‘\check{t}\prime \mathrm{v}_{\theta}*"\check{r}_{P}^{\theta}$$kkx_{\mathrm{k}_{*_{\mathrm{K}_{I_{\mathrm{t}_{\vee}}}}l}}\varphi^{\wedge}.\sim\backslash$$@_{\mathrm{t},\sim},:r^{\{}’\sim$,$\Im_{\sqrt,\mathrm{e}_{\_{r_{\mathit{4}_{\mathrm{a}_{\backslash _{A}}}}}}}$‘$\grave{\sigma\dot{)}\dot{\mathrm{r}}}-\backslash$;$3\mathrm{g}^{b}\mathrm{e}^{d^{r}}$$f^{\wedge},\backslash l_{*,\}}\mathrm{t}*\sim$’$\grave{\mathrm{r}}_{\backslash }\mathrm{R}_{\mathrm{b}}$$r_{P}.\nearrow$:’$d^{\prime^{\chi X}}\check{\wedge}$$j_{\vee r}^{t^{r^{\mathit{3}^{\swarrow}}}}-_{e}$\’u$\sqrt\sim^{r^{\sqrt}}r$$\sim$.

$\cap’ \mathrm{Q}6$$\mathrm{q}|’|$$\eta$$t^{d}.-arrow d^{\sqrt}$:$\hslash_{\mathrm{w}}\nu’\nu_{\ell_{\mu}x\sim^{\wedge}}\prime t$

Figure 10:
into the initial configuration for elongatd for polygons with six comers
and elongation 1.2 (left) md elongation 1.4 (right).
$\mathrm{Z}\infty \mathrm{m}$4.2

Effect of elongation 1.2

For finite elongation, the angle of repose increR for particles with many comers because
the tendency tois suppressed in comparison to non elongaffi partidae. For elongation
1.2 (for a snapshot of the particles, see Fig. 10, left) the angle of repose daeays linearly
ffom 5 to 10 comers, and ffom 10 to 15 corners decays hardly my more. The criticalangles are higher thm for non-elongated particles with the same number of comers.
$\mathrm{r}\mathrm{o}\mathrm{U}$(a) Dependence of the angle of repose on thenumber of comers for polygons with elongation
1.2,
md error-bars calculatd for six
configurations.
$\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}*\mathrm{a}\mathrm{e}$(b) Dependence of the ange of repose for partIclae on the number ofwith elongation
1.4, error-bars for up to 10 configurations.
$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{e}\mathrm{r}8$$\epsilon \mathrm{l}\mathrm{o}\mathrm{p}\mathrm{a}\mathrm{e}/\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}$Figure 11: Dependence of the mge of repose for elongaffi particles with friction coeffi-

cient $\mu=0.6$

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4.3

Effect of elongation 1.4

For larger elongations (elongation 1.4 in Fig. 10, left), in contrast to elongations of 1.2 andsmaller, there is no significant dependence of the critical angle on the number of corners
any more (Fig. ll(b)). That means that for the elongation 1.4, the rolling is alreadyessentially suppressed, the sliding dominates and the particle roughness doae not play a
crucial role in the mobilization of rolling.

Figure 12: Final configuration for coefficient of friction $\mu=0.2$ , particle elongation 1.4
and five corners: Most of the bulk is fluizided, but in the in the middle below the openingof the hopper, a heap with straight slopes is still recognizable.

4.4

Effect of a rough floor

We tried to sI)aed up the simulation by putting particles in a layer on the ground toincrease the enery dissipation, which did not work out well. In the formation of the heapon the lower layer, both sliding of the damping layer and avalanching of the particles
coming from above happen simultaneously. As can be seen in Fig. 13, the slopes are notstraight md the angle of repose is not well defined. As the damping layer constitutes arough ground (albeit a moving one), this result indicates how rough bottoms in simulations
may distort the results.
In Fig. 12, we used elongated particles with a very small $\mu=0.2$ coefficient of friction.
The angle of repose is not measurable from the ground: The particles touch the walls
md form a nearly horizontal layers. Only on this rough layer, a heap forms in the middle
under the hopper which has a measurable angle of repose, due to the rough granular layer
below, not due the particle interactions with the ground82
5Conclusion

In this work we discussed the effect of the shape of particles on the angle of repose, aparadigm on the competition between rolling and sliding in granular materials. We haveshown that the angle of repose for&y granular materials depends crucially on the shapeof the particles and the Coulomb friction coefficient. For non- or moderately elongatedparticles (i.e. up to an elongation 1.2), we found a strong dependence of the angle of repose
on the particle roughness, i.e. for our polygonal particles the angle of reposed decreased
with increasing number of corners. This result is in marked contrast to the results fortriaxial compression, where one finds hardly any dependence of the stress-strain-diagramon the particle roughness [8]. For larger elongations, the effect of the particle roughness
on the angle of repose is suppressed, because no rolling takes place anyway. In this case,the angle of repose is mostly dependent on the coefficient of friction.
Simulations which neglect even only one of these influences will not be able to reproduce neither the statics (angle of repose, stress distribution) nor the dynamics (e.g.
avalanches, force networks) for realistic materials. Due to the tw0-dimensional simulation,
the fluctuations on the data are nevertheless more pronounced than in three-dimensional
experiments. The whole mechanism of fomation of heaps and angle of repose is the resultof the particle properties (shape and inter-particle friction) on the one hand, and of sliding
on the ground and avalanches on the other hand.

References[1] J.A. C. Gallas, S. Sokolowski, Grain non-sphericity effects on the angle of repose ofgranular material, Int. Journ. Mod. Phys 7, 2037 (1993).
[2] T.P\"oschel, V. Buchholtz, Static friction phenomena in granular materials: Coulomb
law versus particle geometry, Phys. Rev. Lett. 71, 3963 (1993).

[3] J. Lee, H. J. Herrmann, Angle of repose and angle of marginal stability: moleculardynamics of granular particles. J. Phys. A: Math. Gen 26, 373 (1993).
[4] H.-G. Matuttis, Simulation of the pressure distribution under a tw0-dimensional heapof polygonal particles, Granular Matter 1, p. 83-91 (1998)[5] A. L. Barabasi, R. Albert, P. Schiffer, The physics of sand castles: maximum angleof stability in wet and dry granular media, Physica A 266366371 (1999).

[6] A. Schinner and H.-G. Mattutis, Molecular Dynamics of Cohesive Granular Materials,
in: Traffic md Granular Flow, e&. D. Helbing, H.J. Hermann,[7] H.-G. Matuttis, A. Schinner, Particle Simulation for Cohesive Granular Materials,
Int. Journ. Mod. Phys. C, Vol12, Nr. 7, p. 1011-1022.

[8] H.-G. Matuttis, Nobuyasu Ito, Alexander Schinner, Effect of particle shape on bulkstress-strain relations of granular materials, Proceedings of RIMS Symposium on
Mathematical Aspects of Complex Fluids III, RIMS Kokyoroku series 1305, p.89-99(2003), and: S. El. Shourbagy, Sh. Morita, H.-G. Matuttis, in preparatio

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Figure 13: Initial state (above)and end configuration (below) of a simulation for thecomputation of the outflow from a hopper on a rough ground with friction coefficient$\mu=0.6$ onto a layer with particles for increased enery damping. The slopes are notstraight due to the superposition of the out-flowing grains md the damping layer. Particles
in the same layer have the same shade of gray, mixing of shades in the end configurationof layer.
indicates$\mathrm{m}\ddot{\mathrm{m}}\mathrm{n}\mathrm{g}$