TitleEffect of particle shape on bulk-stress-strain relations ofgranular materials (Mathematical Aspects of Complex FluidsIII)

Author(s) Mattutis, Hans-Georg; Ito, Nobuyasu; Schinner, Alexander

Citation 数理解析研究所講究録 (2003), 1305: 89-99

Issue Date 2003-02

URL http://hdl.handle.net/2433/42784

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

Effect of particle shape on bulk-stress-strain relations of granularmaterials

Hans-Georg Mattutis, University of Electro-Communications, Department of Mechan-ical and Control Engineering, Chofu, Chofugaoka 1-5-1, Tokyo 182-8585, JapanNobuyasu Ito, Department of Applied Physics, School of Engineering,The University ofTokyo, BunkyO-ku, Tokyo 113, JapanAlexander Schinner, GeNUA mbh, R\"aterstrffie 26, 85551 Kirchheim, Germany

AbstractThe effect of the particle shape on the bulk-stress-strain-relations for triax-ial compression of granular media is investigated via the molecular dynamicsmethod. It is found that crucial properties exhibited by experimental granularmedia cannot be reproduced by round particle simulations, but only by theuse of elongated particles.

1IntroductionThe paradigm of statistical physics is toobtain the description of macroscopic Phe- $\hat{3}$nomena as statistical effects of the micr0- $\mathrm{a}$scopic compounds. In solid state physics, $\triangleleft \mathrm{q}$)

one is used to the fact that replacing on $.\underline{\mathrm{e}}$.kind of atoms in asolid alters the macr0- $4$)$\infty$scopic properties of the solid, e.g. dop- $\underline{8l\approx \mathrm{O}}$ing improves electric conductivity in semi- $4\dot{0}$conductors, adding chromium and man- $\frac{\mathrm{q})}{\mathrm{b}}0$ganese increase the hardness of steel etc. 5“Round” noble gas atoms have low evap0-ration temperatures, whereas solids madeffom atoms with “rough” covalent bondsexhibit high stability. In the granular com-munity, most particle simulations are still Figure 1: Angle of repose for aheap fromperformed using round particles, the ”par- mon0- isperse particles with different num-ticle 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 as-semblies.

2Shape effects in granular researchIn this section, give abrief overview about some established particle shape effects ingranular materials

数理解析研究所講究録 1305巻 2003年 89-99

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

3.1 Stress-Strain relations for experimental granular materialsAcharacteristic 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

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$arrow$

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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 themaximal 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 compressionwith polygonal particles. The increase in the pore space due to Reynolds dilatancy duringcompression is clearly visible. One can see that the deformation of the sample volume isinhomogeneous.

3.2 Setup of the simulationIn our two dimensional simulation, triaxial compression reduces of course to biaxial com-pression, where the system is held under constant pressure along the $\mathrm{x}$-axis and com-pressed under constant velocity along the $\mathrm{y}$-axis. Because the simulation of the rubbermembrane and the surrounding water would create an unnecessary overhead, we simulatethe boundary of the axis which is not the compression axis via friction-less hard wallswhich are held by anon-linear spring which tries to keep the pressure constant, see Fig. 2.The modulus of elasticity of the particles was $10^{6}\mathrm{N}/\mathrm{m}$, the normal damping was ch0-sen so that the tw0-particle collision was nearly “totally inelastic”. All simulations were

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performed with the same normal dissipation. The coefficient for the Coulomb frictionwas chosen as 0.6 if not mentioned otherwise. The force models used for the ellipses aredescribed in[6], for the polygons in[7]. The external pressure was kept at $10^{4}\mathrm{N}/\mathrm{m}$ . Avertical stress of 15 $\cdot 10^{4}$ translates therefore into 15 times the horizontal stress. The vol-umes are given so that the volume at the beginning of the compression, when the upperlid 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 avaryingnumber of corners. The proportion of the longer to the shorter half axis of the ellipseswill be in the following be denoted as “elongation”. In the following, all strains are linearstrains, the height of the volume is rescaled by the initial height.

$\Re\frac{\Phi}{\tilde,\Phi}$

,

Figure 6: Stress-strain diagram for asin- Figure 7: Density-strain diagram for thegle run of mon0-disperse round particles runs in Fig. 6for mon0-disperse round(full line), polydisperse round particles particles (full line), polydisperse round(dashed line) and polydisperse elongated particles (dashed line) and polydisperseparticles (dotted line), scaled by the ex- elongated particles (dotted line), scaled bytemal pressure. the density before compression.

4Stress-Strain relations

4.1 Results for single runsFor the sake of clarity, we will compare curves made for single runs of mon0-disperseround particles, polydisperse round particles and polydisperse elongated particles. Thestress-strain and density-strain relations for granular materials differ crucially dependingon the shape of the particles. An aggregate of elongated particles with average elongation1.8 can carry nearly twice as much stress as round polydisperse and mon0-disperse par-ticles, see Fig. 6. Moreover, the stress-strain distribution for “usual” granular media hasamaximum, acriterion which is only fulfilled for elongated particles. For the density-strain-distribution, it is surprising that the maximal density for elongated polydisperse

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

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

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

Figure 9: Stress-strain diagram for ellipses Figure 10: Stress-strain diagram for el-of average elongation 1.8. lipses of average elongationl.l.

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

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

4.3 Elliptic particlesAs 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 halfthe 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 hasto 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 and $\infty$ number ofcorners

4.4 Roughness of the particlesWhereas the effect of the elongation turnedout to be considerable, the effect of the rough-ness of the particles was not so crucial. Poly-gons have a“rougher” appearance than el-lipses, and polygons with fewer corners lookrougher than those with more corners. Fig. 12shows the stress-strain diagram for polygonswith 7corners for the same average length-and size istribution as for the ellipses inFig.9. The only difference in the stress-strainrelations of ellipses and polygons with 7cor-ners 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 as-ing, because intuitively, one associates higher semblies of polygonal particles of averageshear resistivity with rougher surfaces. It elongation 1.8 and 7 corners.should be noted, that our investigation of “roughness” does not compare truly differentsurfaces, but particles with smooth surfaces, only adifferent number of corners. Usingpolygons 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 parti-cles with elongation 1.8 and vanishingCoulomb-friction.

Figure 14: Stress-strain curve for parti-cles with elongation 1.8 and Coulomb-friction coefficient 0.3.

4.5 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 frictioncoefficient with $\mu=0.6$ in Fig. 12 and Fig. 9, though the size dispersion and the particleelongation were the same. For the friction coefficient $\mu=0.3$ , for the given averageelongation of 1.8 the maximum in the shear stress which should be characteristic forgranular media is hardly visible, and the corresponding relative minimum in the volume isonly 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 forgranular 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 materialsand liquids will occur if the size of the granular materials will be reduced, and at whichlength-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 alength-scale $

Figure 15: Ellipses with elongation 1.44and 32 corners.

Figure 16: Stress-strain and stress-densitydiagram 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 simula-tions 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

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civil engineering, the pressure distribution in the middle of the heap remains flat. Via com-puter simulations, we could identify the reason for the “dip” as the density inhomogeneityresulting 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 alayeredsequence 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 densitypattern for filling from apoint source, and the density inhomogeneity was also found tobe up to about 5%. In our preceding research we were astonished that groups using sim-ulations 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 distributionsExperimentally, it has been found that heaps made from non-cohesive non-elongated par-ticles show no (glass beads $>0.5mm$ diameter[12]) or only avanishing (rape seed, [13], ar-guably slightly elongated) pressure minimum even if they are piled up in awedge sequence,whereas small, cohesive glass beads ($

Figure 18: Density of aheap built from apoint source. The high-density cone in themiddle 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 irregularand 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 adensityvariation 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 piledin 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 necessarilyapplicable to aggregates made of elongated particles.

Relating granular particles and granular bulk properties, we have shown that bulkproperties 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 fordilute systems (”single-particle-problem”)become important in the dense regime, whereother properties which are important in the dilute regime, like normal dissipation, become

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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 InternationalConference on Discrete Element Methods (DEM), edited by J. R. Williams (IESLPubl., 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.

[6] H.-G. Matuttis, N. Ito, H. Watanabe, and $\mathrm{K}.\backslash$ M. Aoki, in Powders b Grains 2001,edited by Y.Kishino (Balkema, Rotterdam, 2001), pp. 173-176.

[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, andK. Kassner, in Powders&Grains 2001, edited by Y.Kishino (Balkema, Rotterdam,2001), pp. 499-502.

[11] J. $\check{\mathrm{S}}$mid, P. V. Xuan, and J. Thyn, Chem. Eng. Technol. 16, 114 (1993).

[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, andK. Kassner, in Mathematical Aspects of Complex Fluids II, Vol. 1184 of Kokyuroku(Research Insitute for Mathematical Sciences, Kyoto University, 2001), pp. 123-139

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