数理解析研究所講究録 1305 巻 2003 年 89-99

89
Effect of particle shape on bulk-stress-strain relations of granularmaterials
Hans-Georg Mattutis, University of Electro-Communications, Department of Mechanical and Control Engineering, Chofu, Chofugaoka 1-5-1, Tokyo 182-8585, Japan
Nobuyasu Ito, Department of Applied Physics, School of Engineering , The University ofTokyo, BunkyO-ku, Tokyo 113, Japan
Alexander Schinner, GeNUA mbh, R\"aterstrffie 26, 85551 Kirchheim, Germany
Abstract
The effect of the particle shape on the bulk-stress-strain-relations for triaxial compression of granular media is investigated via the molecular dynamics
method. It is found that crucial properties exhibited by experimental granular
media cannot be reproduced by round particle simulations, but only by theuse of elongated particles.

1Introduction
The paradigm of statistical physics is toobtain the description of macroscopic Phenomena as statistical effects of the micr0scopic compounds. In solid state physics,
one is used to the fact that replacing on)
kind of atoms in asolid alters the macr0scopic properties of the solid, e.g. doping improves electric conductivity in semiconductors, adding chromium and manganese increase the hardness of steel etc.
“Round” noble gas atoms have low evap0ration temperatures, whereas solids madeffom atoms with “rough” covalent bonds
exhibit high stability. In the granular community, most particle simulations are still Figure 1: Angle of repose for aheap fromperformed using round particles, the ”par- mon0- isperse particles with different numticle shape” as amaterial parameter is still bers of corners. After [1],widely neglected. In this paper, we want to show how the macroscopic stability is affectedby the particle shape, and how the stability influences further properties of granular assemblies.
$\hat{3}$$\mathrm{a}$.
)$\triangleleft \mathrm{q}$$.\underline{\mathrm{e}}$$4$$\infty$$\underline{8l\approx \mathrm{O}}$$4\dot{0}$$\frac{\mathrm{q})}{\mathrm{b}}0$52Shape effects in granular research
In this section, give abrief overview about some established particle shape effects in
granular materials

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3Triaxial Compression
The standard test of granular materials to derive their stress-strain curve is the triaxial
compression, in which the compression along the -axis occurs with constant velocity,whereas the pressure along the $x-$ and -axis is held constant. Experimentally, the sidesof the granular medium are held by arubber membrane, the constant pressure is realized
by the constant pressure of awater reservoir on the rubber membranes, see Fig. 2.
$z$

$y$

3.1

Stress-Strain relations for experimental granular materials

Acharacteristic of the stress-strain relations for granular media under triaxial compressionin contrast to e.g. uniaxial compression of metals is, that the stress curve has aclea

91$arrow$$arrow$

Figure 4: Schematic drawing for the triaxial compression (left) and the stress-strain curvefor granular materials under triaxial compression (right), after [5]maximum, where the material is in aplastic regime, for larger stresses the material entersthe failure regime where the stress necessary to deform the material is smaller than the
maximal yield stress, and the density-maximum/minimum of the volume is not reachedat the maximum stress, but before, due to Reynolds dilatancy, see Fig. 4.

Figure 5: Initial, intermediate and final stage of asimulation run for triaxial compression
with polygonal particles. The increase in the pore space due to Reynolds dilatancy during
compression is clearly visible. One can see that the deformation of the sample volume isinhomogeneous.

3.2

Setup of the simulation

In our two dimensional simulation, triaxial compression reduces of course to biaxial compression, where the system is held under constant pressure along the -axis and compressed under constant velocity along the -axis. Because the simulation of the rubber
membrane and the surrounding water would create an unnecessary overhead, we simulate
the boundary of the axis which is not the compression axis via friction-less hard walls
which are held by anon-linear spring which tries to keep the pressure constant, see Fig. 2.
, the normal damping was ch0The modulus of elasticity of the particles wassen so that the tw0-particle collision was nearly “totally inelastic”. All simulations were$\mathrm{x}$$\mathrm{y}$$10^{6}\mathrm{N}/\mathrm{m}$92performed with the same normal dissipation. The coefficient for the Coulomb friction
was chosen as 0.6 if not mentioned otherwise. The force models used for the ellipses are. A
described in[6], for the polygons in[7]. The external pressure was kept attranslates therefore into 15 times the horizontal stress. The volvertical stress of 15
umes are given so that the volume at the beginning of the compression, when the upper
lid gets in contact with the granular filling (left picture in Fig. 5), is normalized to 1.
The particles used were either ellipses or polygons inscribed into ellipses with avarying
number of corners. The proportion of the longer to the shorter half axis of the ellipses
will be in the following be denoted as “elongation”. In the following, all strains are linear
strains, the height of the volume is rescaled by the initial height.
$10^{4}\mathrm{N}/\mathrm{m}$$\cdot 10^{4}$$\Re\frac{\Phi}{\tilde,\Phi}$,

Figure 6: Stress-strain diagram for asingle run of mon0-disperse round particles(full line), polydisperse round particles(dashed line) and polydisperse elongatedparticles (dotted line), scaled by the extemal pressure.

Figure 7: Density-strain diagram for theruns in Fig. 6for mon0-disperse roundparticles (full line), polydisperse roundparticles (dashed line) and polydisperse
elongated particles (dotted line), scaled by
the density before compression.

4Stress-Strain relations
4.1

Results for single runs

For the sake of clarity, we will compare curves made for single runs of mon0-disperse
round particles, polydisperse round particles and polydisperse elongated particles. The
stress-strain and density-strain relations for granular materials differ crucially dependingon the shape of the particles. An aggregate of elongated particles with average elongation
1.8 can carry nearly twice as much stress as round polydisperse and mon0-disperse particles, see Fig. 6. Moreover, the stress-strain distribution for “usual” granular media has
amaximum, acriterion which is only fulfilled for elongated particles. For the densitystrain-distribution, it is surprising that the maximal density for elongated polydisperse

93$\frac{\omega\circ\emptyset)arrow}{\omega}$$=w^{1}uw\circ$

Figure 8: Raw data for four runs of stress-strain-curves (left) and the same data realigned
along the slope (right) .

can be up to 5% and for mono disperse round particles up to 5%higher than for polydisperse round particles. As can be seen in Fig. 6, if the compression of elongated particles
is continued, the stresses rise again after the failure regime.

Figure 9: Stress-strain diagram for ellipsesof average elongation 1.8.

4.2

Figure 10: Stress-strain diagram for ellipses of average elongationl.l.

The misery of averaging noisy curves

Especially two dimensions, the simulations data suffer from strong fluctuations. Averagesof curves with relative maxima do not necessarily yield acurve with amaximum. Because
the maximum of the single curves is of different size at different locations, the averaging
may actually cancel out the maximum. To reduce this effect, we aligned the slope of the

94stress-strain curves, see Fig. 8. In all following stress-strain diagrams, the statistic error
bars of the 4to 6samples are indicated-signs in the graphs.
$\mathrm{b}\mathrm{y}+$4.3

Elliptic particles

As for the single runs, the average maximal yield stress for ellipses (average elongation 1.8,
Fig. 9) is nearly twice as large as for circles of the same size dispersion (Fig. 10). Comparedto aggregates of elongated particles, aggregates of round particles have nearly only half
the stability, as far as the yield stress is concerned, and also the relative maximum of theshear stress was conspicuously absent. The minimum for the volume of the cell (maximumof the density) is also markedly shallower for round than for elongated particles. One has
to conclude that round particles do not exhibit proper behavior of granular materials intheir stress-strain characteristics. We found that it was possible to manipulate the setupof the compression in such away that amaximum in the stresses appeared by fixing oneof the walls. Such asetup is definitely different ff\‘Om the true experimental setup.

Figure 11: “Elliptic-shaped” particles of elongation 1.8 with 7,14,32 andcorners

4.4$\infty$number of

Roughness of the particles

Whereas the effect of the elongation turned
out to be considerable, the effect of the roughness of the particles was not so crucial. Polygons have a“rougher” appearance than ellipses, and polygons with fewer corners lookrougher than those with more corners. Fig. 12shows the stress-strain diagram for polygons
with 7corners for the same average lengthand size istribution as for the ellipses in
Fig.9. The only difference in the stress-strain
relations of ellipses and polygons with 7corners that the fluctuations are increased in thecase of polygons, within the fluctuations, theposition of the maximum and its numericalvalue is the same. This is rather surpris- Figure 12: Stress-strain diagram for asing, because intuitively, one associates higher semblies of polygonal particles of average
shear resistivity with rougher surfaces. It elongation 1.8 and 7 corners.
should be noted, that our investigation of “roughness” does not compare truly different
surfaces, but particles with smooth surfaces, only adifferent number of corners. Using
polygons instead of ellipses does not alter the macroscopic stress-strain behavior, and fortruly modeling “micr0-roughness”, using polygons with afew corners is not sufficient.

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Figure 13: Stress-strain curve for particles with elongation 1.8 and vanishing
Coulomb-friction.

4.5

Figure 14: Stress-strain curve for particles with elongation 1.8 and Coulombfriction coefficient 0.3.

Friction effectEllipses with an elongation of 1.8 approximated by polygons with 32 corners with $\mu=0.0$(left) and $\mu=0.3$ (right) show adefinite deviation ffom the simulations with friction
coefficient with $\mu=0.6$ in Fig. 12 and Fig. 9, though the size dispersion and the particle
elongation were the same. For the friction coefficient $\mu=0.3$ , for the given average
elongation of 1.8 the maximum in the shear stress which should be characteristic for
granular media is hardly visible, and the corresponding relative minimum in the volume is
only very weak. For the same size- and shape-distribution at vanishing friction coefficient$\mu=0.0$ , neither stress-strain nor stress-volume curves resemble those for typical for
granular materials, and one has to conclude that simulations of granular materials atvanishing Coulomb friction do not realy offer insights into realistic granular materials.
In this context, it is appropriate to ask where the transition between granular materials
and liquids will occur if the size of the granular materials will be reduced, and at which
length-scale the granular medium will become aliquid. Current research in the fieldof friction in nanotribology[8] shows that Coulomb friction exists on surfaces up to a
. Therefore, apowder made of solid particles, where the atoms have alength-scale
fixed neighborhood relation can be expected to behave like agranular medium for agrain
size above $5nm$ , albeit acohesive one. In contrast, those polymer molecules which havenot afixed neighborhood relation show in nan0-scale experiments no Coulomb friction
characteristics, and they will therefore also not form agranular material even if the sizeof the molecules is considerably above $5nm$ .
$<5\mathrm{n}\mathrm{r}\mathrm{a}$4,6Elongation effect for polygons

As in the case of ellipses, also for polygons the maximal shear stress is also decreased incomparison to more elongated particles, be it ellipses or polygons. In Fig. 15, the particles

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Figure 15: Ellipses with elongation 1.44
and 32 corners.

Figure 16: Stress-strain and stress-density
diagram for mon0-disperse polygons with32 corners and an elongation of 1.8.

with an elongation 1.44 and 32 corners have amaximal shear stress which is between theellipses of elongation 1.8 in Fig. 9. and the circles of Fig. 10.

In aprevious study, we found via simulations that the pressure distribution under sand heaps depends on the history[7], aresultthat was later found experimentally[9]. For a“wedge sequenc\"e, the construction methodfavored in powder technology, the pressure under heaps for non-spherical particles yieldpressure minima (a “dip”), for alayered sequences, the construction history favored in97
civil engineering, the pressure distribution in the middle of the heap remains flat. Via computer simulations, we could identify the reason for the “dip” as the density inhomogeneity
resulting from the filling via the point source[10]. Such aheap density is reproduced inFig. 18, with adensity variation of about 5%. The density of aheap build in alayered
sequence is given in Fig. 19. We could not find experimental density investigations forsand heaps, but the density measurements for on silo fillings[ll] indicate the same density
pattern for filling from apoint source, and the density inhomogeneity was also found to
be up to about 5%. In our preceding research we were astonished that groups using simulations of round particles were not able to reproduce our results, inhomogeneous densityand pressure dip, which we obtained for elongated particles. The reason to undertakethe present study was to clear up the relation between particle shape, stress-strain andstress-density distribution.

5.2

Influence on the particle-shape on the pressure distributions

Experimentally, it has been found that heaps made from non-cohesive non-elongated particles show no (glass beads $>0.5mm$ diameter[12]) or only avanishing (rape seed, [13], arguably slightly elongated) pressure minimum even if they are piled up in awedge sequence,
whereas small, cohesive glass beads ( $<0.5mm$ diameter) and sand in the same experiments showed amarked pressure minimum. The appearant violation of Bagnold-scaling,where the pressure minimum depends on the size of the glass beads, can be resolved byreinterpreting cohesive non-elongated particles as elongated polydisperse particles, likemolecules made up from atoms. Our simulations showed that indeed the pressure minimum developed for non-elongated particles with increasing inter-particle cohesion[14].
The reason why polydisperse round particles showed no pressure dip can be understood if
the stress-strain and density-strain relations for elongated and non-elongated particles arecompared. If one accepts that the pressure minimum under aheap is due to the density
inhomogeneity, which allows the formation of an arch over the high-density cone in the
middle of the heap, then the variation in the bulk density of up to 5%density differenceis anecessary criterion for the formation of adip. In our triaxiaJ compression simulations, the density variation from the uncompressed sample up to the maximal density forelongated particles is also up to 3-5 %. In contrast, for non-elongated particles, especiallyin the absence of static friction, the density-strain curve allows no sufficient variation of
the density. Moreover, the stability of such heaps is smaller than that of heaps formed
from elongated particles, so that existing density inhomogeneities are much more easily
extinguished by macroscopic reordering during the pile-up process.

6Conclusion:
Afundamental conlusion of our study is, that aggregates of elongated particles and particles with friction are much more stable than aggregates of round particles or particles
without friction. For particulate modeling in dense granular materials, the particle shape
has to be considered as fundamental parameter, like the friction coefficient. Only for realistic particle shapes and realistic coefficients of friction, the “true” stress-strain behaviorof granular materials can be recovered

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Figure 18: Density of aheap built from apoint source. The high-density cone in the
middle leads to the arching which caused the pressure dip.

Figure 19: Density of aheap built from alayered sequence. Due to the homogeneousdensity, the pressure on the base is constant under the apex.
The outcome of experiments with glass beads which do not resemble the irregular
and usually elongated shape of conventional granular materials can according to oursimulations be expected to show adifferent macroscopic behavior ffom that of “true”
granular materials even in the most fundamental mechanical property, the mechanicalstrength. The value of studies of round particles for the understanding of realistic granularmaterials is therefore problematic.
For the formation of apressure dip under heap, it is necessary to to maintain adensity
variation within the heap. This, and ahigher bulk shear strength can only be obtainedwith elongated particles. Adensity variation of 5%is must be considered significant forRayleigh dilatancy and the pressure dip under heaps. Density measurements for granularmaterials piled e.g. in alayered sequence are not meaningful if the same material is piled
in awedge sequence.
For elongated particles, the stress-strain curve is not monotonous, therefore it is notpossible to determine the stresses in granular aggregates from the corresponding densities.
For non-elongated particles, the stress-strain curve is monotonous uP to the plastic regime,
but findings for granular aggregates made of non-elongated particles are not necessarily
applicable to aggregates made of elongated particles.
Relating granular particles and granular bulk properties, we have shown that bulk
properties of granular materials like the relative minimum of the stress-strain curve canonly be obtained for elongated particles.
Some single particle properties (shape, friction coefficient) which are unimportant for
dilute systems (”single-particle-problem”)become important in the dense regime, whereother properties which are important in the dilute regime, like normal dissipation, become

99
unimportant, because no dynamic collisions occur.

References[1] A. Schinner, Ph.D. thesis, University of Magdeburg, 2000.
[2] L. M. John M. Ting, Jeffrey D. Rowell, in Proceedings on the 2nd International
Conference on Discrete Element Methods (DEM), edited by J. R. Williams (IESL
Publ., Cambridge, Mass, 1993), pp. 215-225.

[3] F. Radjai, M. Jean, J. J. Moreau, and S. Roux, Phys. Rev. Lett. 77, 274 (1996).
[4] P. G. de Gennes, Talk in the Groupement de Recherche meeting for of dry granularMedia, 22.01.1999, Paris, Prance.

[5] after Stefan Diebels, personal communication.

M. Aoki, in Powders b Grains 2001,[6] H.-G. Matuttis, N. Ito, H. Watanabe, andedited by Y.Kishino (Balkema, Rotterdam, 2001), pp. 173-176.
$\mathrm{K}.\backslash$[7] H.-G. Matuttis, Granular Matter 1, 83 (1998).
[8] ISSP-Workshop on Physics of Friction, 23.10-25.10.2002.
[9] L. Vanel, D. Howell, D. Clark, R. P. Behringer, and E. Clement, Phys. Rev. E60,
R5040 (1999).
[10] A. Schinner, H.-G. Mattutis, J. Aoki, S. Takahashi, K. M. Aoki, T. Akiyama, and
K. Kassner, in Powders&Grains 2001, edited by Y.Kishino (Balkema, Rotterdam,
2001), pp. 499-502.

[11] J. mid, P. V. Xuan, and J. Thyn, Chem. Eng. Technol. 16, 114 (1993).
$\check{\mathrm{S}}$[12] R. Brockbank, J. M. Huntley, and R. Ball, J. Phys. II France 7, 1521 (1997).
[13] T. Jotaki and R. Moriyama, Journal of the Society of Powder Technology, Japan 16,
184 (1979).

[14] A. Schinner, H.-G. Mattutis, J. Aoki, S. Takahashi, K. M. Aoki, T. Akiyama, and
K. Kassner, in Mathematical Aspects of Complex Fluids II, Vol. 1184 of Kokyuroku(Research Insitute for Mathematical Sciences, Kyoto University, 2001), pp. 123-139