数理解析研究所講究録
1184 巻 2001 年 123-139

123

Towards a micromechanic understanding of the pressure
distribution under heaps
Hans-Georg Mattutis1 , Alexander Schinner2 , Tetsuo Akiyama4 , Junya Aoki4, Satoshi
Takahashi4, Keiko M. Aoki3, Klaus Kassner2
1
Department of Applied Physics, School of Engineering,The University of Tokyo, Bunkyoku, Tokyo 113, Japan
2
Otto-von-Guericke-University Magdeburg, Universit\"atsplatz 2,D-390l6 Magdeburg, Germany
3 JST,
nano-LC project, TRC, Tokodai 5-9-9, Tsukuba 300-2635, JAPAN4 Departmentof Chemical Engineering, Shizuoka University, Hamamatsu, 432, Japan
Abstract
The pressure distribution under heaps has found to be dependent on the building history of the heap both in experiments and in simulations. Up to now,theoretical models and analysis assume that the packing of the heap is homogeneous. We show new experimental and simulational results which indicatethat the packing is inhomogeneous and that this packing property is likely
causing the pressure minimum under the heap.

1

Introduction

124
1.1

Arching

The first arches which wherebuilt into a ”granular heap”
where the tomb chambers in
Egyptian pyramids. The pyramids internal structure, seeFig. 2, reminds to the stressmodel given in Ref. [2]. After ”false arches” in Egypt andancient Greece, ”true arches”
were discovered as a principle
in civil engineering by the Romans, and with little modifications as Gothic arches, the Figure 2: Crossection of the pyramid of Pharao Sahure) at Abu Sir. The building
concept is prevalent till today. (dated about 2500
Arching as a topic in granu- method is not homogeneous, but consists of horizonlar materials is a comparatively tal masonry, inclined walls to direct the pressure out of
new concept, dating back not the middle, and large blocks which form a”false arch””
much more than hundred years. above the tomb chamber. Usually, the angle of repose”
The ffist references on a mathe- in pyramids is 56 degrees.
matical treatment of” Bogenbildung” (arching) in Terzaghi’s textbook on ”Erdbaumechanik” (geotechnics) [13] go back
. Arching as a”prime suspect” forto the last quarter of the nineteenth
the formation of the pressure dip was already mentioned in the experimental papers on.
the pressure$\mathrm{B}.\mathrm{C}.$$\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{y}[14,15]$$\mathrm{d}\mathrm{i}\mathrm{p}[6]$1.2

Phenomenology of arching under granular heaps

125
showed nearly scalable results for sand and granular fertilizer for different heap sizes,
but as both materials are composed of rough, polydisperse grains, this could rather be
expected. Ref. [8] used the widest range of different materials for cones built in a wedge
sequence, ranging from sand over small and large monodisperse glassbeads. Polydispersematerials turned out to always yield a pressure minimum. Monodisperse smooth large
glass beads exhibited no pressure minimum, but small monodisperse and frosted large
glass beads showed a pressure dip. This was a crucial finding, as the size of the particles
and their surface properties seemed to influence the resulting pressure distribution. The
small monodisperse glass beads were of a size ( $<0.4$ mm) so one had to suspect effectsof cohesion. Finally, the effect of the construction history was verified experimentallyin Ref.[12], where wedge sequences showed a pressure dip in wedges and cones, whereas
layered sequences showed a pressure dip neither in wedges nor in cones.

Figure 4: “Generic” models for the pressure clistribution under a heap in a) to c) and”model in d), e).
of the angle of repose in a‘($\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{l}\cdot \mathrm{i}\mathrm{c}$$\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{p}\iota 1\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$Why we don’t use ”theory” (yet)1.3.

of the grains seem to influence the forming of a heap,
or from using simple computationalofhave to ) absorbed in a single parametcr. In ptysicalunderstanding of physicalforarclno(lelLb forclistributions witl]
canFig. 4. Tlle generic moclclsof th( heap,
in thepressnre lllaxilna or flatgrains (”ofcomposccl
oftodiscret,e)(as ill Fig. ) or
Ref. ]).
fielel
])
.
)
(
](
. In
.
.
of
of {

Bccause many physicalthe
we refrained
modcls, where several (scienc , gencri($\mathrm{f}\mathrm{r}\mathrm{o}\ln$$\mathrm{P}^{\mathrm{r}\mathrm{o}}\mathrm{I}^{)\mathrm{e}1\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{s}}$$11,\mathrm{s}\mathrm{e}$$\mathrm{m}\mathrm{a}\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{c}\cdot o\mathrm{p}\mathrm{i}\mathrm{e}\cdot \mathrm{m}\mathrm{o}(\mathrm{l}\mathrm{c}\mathrm{l}\mathrm{s}$$[$$\mathrm{c}$$111\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{c}^{1}\mathrm{s}$

”

”

$.$

$\mathrm{t}\mathrm{h}\mathrm{t}\mathrm{l},$$1\mathrm{n}\mathrm{o}\mathrm{d}(^{\backslash }1\mathrm{s}$$\mathrm{f}\mathrm{l}\iota \mathrm{n}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{l}$$(^{1}\mathrm{s}$$\iota\iota \mathrm{s}\iota \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}_{1},,\iota\llcorner \mathrm{s}^{\mathrm{I}}\mathrm{c}\mathrm{c}\mathrm{l}$”$\mathrm{t}_{)(^{1\{}}^{\mathrm{T}}‘ 1\iota \mathrm{p}$$\mathrm{P}^{1\mathrm{o}\mathrm{c}\mathrm{e}^{\mathrm{c}}\mathrm{i}\mathrm{S}(^{)}\mathrm{S}}.‘\cdot()\mathrm{f}(.0\iota\iota \mathrm{r}_{\iota}^{\zeta^{1}},\mathrm{c}()11\mathrm{C}^{1}$$\mathrm{t}\mathrm{h}\mathrm{c}_{\mathrm{I}^{)\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{t}^{\tau}}}$$\mathrm{g}_{\mathrm{C}^{1}11}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{c}$$11\iota \mathrm{i}\mathrm{d}\mathrm{d}\mathrm{l}\mathrm{c}$$\backslash$$\mathrm{s}(^{\mathrm{J}}\mathrm{c}$$\mathrm{c}\mathrm{u}\mathrm{r}T^{r}\mathrm{C}^{1}\mathrm{S}$$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{c}\mathrm{s}_{1^{)()11\mathrm{d}\mathrm{t}1_{1\mathrm{C}11}}}$)$\mathrm{i}\mathrm{l}\rfloor$$111\mathrm{C}_{C}^{1}‘ 111$$\mathrm{r}\mathrm{c}_{\mathrm{P}^{1(_{}^{\backslash }\mathrm{b}(^{1}111\mathrm{a}\mathrm{f}\mathrm{i}\mathrm{v}\mathrm{t}^{1}\mathrm{v}\mathrm{o}1_{11111\mathrm{t}^{\tau}\mathrm{b}^{\backslash }}}}\cdot,$$\iota \mathrm{r}\mathrm{l}\subset‘ \mathrm{l}\mathrm{n}\mathrm{y}$‘$\mathrm{i}\mathrm{l}\iota \mathrm{g}$$\mathrm{c}\gamma \mathrm{v}\mathrm{e}^{1}1^{\cdot}\mathrm{c}\gamma \mathrm{b}$”$\mathrm{r}()\mathrm{x}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{l}$$\mathrm{W}1_{1(^{3}}\mathrm{f}\mathrm{h}\mathrm{C}^{\backslash }\mathrm{r}\mathrm{o}11(^{\backslash }1\mathfrak{i}^{\backslash },(^{\mathrm{Y}\mathrm{L}}.,,$$\lceil 2$$\mathrm{a}\mathrm{p}$$- 4$$\mathrm{d}\mathrm{c}\mathrm{l}$$\mathrm{m}\langle$

$‘\}$
(

$\uparrow]_{1\mathrm{e}^{\mathrm{Y}}\mathrm{t})11\mathrm{t}(()\rceil \mathrm{J}\mathrm{l}\mathrm{t}^{\backslash }}$$\dagger$$\mathrm{o}\mathrm{f}\mathrm{h}\mathrm{t}_{\zeta}’.1_{\mathrm{u}}\mathrm{S}\mathrm{t}^{\mathrm{Y}}\mathrm{S},$$1\mathrm{i}\mathrm{H}^{\cdot}(^{1}\mathrm{r}(^{i}1)\uparrow \mathrm{i}c‘ 11(^{i}(1^{\iota\iota_{\mathrm{c}}\iota \mathrm{t}\mathrm{i}\mathrm{t})11\mathrm{h}(]_{()(_{1)}^{\backslash \mathrm{C}^{\backslash }}11()}\mathrm{t}}‘,’ 1\mathrm{t}1i\mathrm{t}\mathrm{t}\zeta(^{1}\mathrm{n}\downarrow 1\rceil(.11(1\mathrm{i}14^{\cdot}\mathrm{t}^{\backslash }1^{\cdot}\mathrm{t}’ \mathrm{l}1(.$$]^{)_{\dot{\zeta}}\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{i}}(\gamma$$.$$]\mathrm{l}\langle^{\backslash }(’ \mathrm{c}‘ 11(.\iota\iota 1\mathrm{c}‘ \mathfrak{i}1\mathrm{i}()11]_{11\mathrm{h}^{1}}\dot{‘}‘|)(^{\backslash }\langle^{1}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{l}111^{\cdot}[]1\downarrow \mathrm{h}^{\tau}\mathrm{i}(_{(}..|11\backslash \cdot(]_{()}\{(^{\mathrm{Y}}\mathrm{l}.\mathrm{J}1\downarrow \mathrm{i}11\mathrm{t}^{1}(11)\backslash ’ 11_{1\mathrm{t}^{\mathrm{Y}}(}\cdot 1_{1()}\mathrm{i}\mathrm{r}\cdot()$]$.,])1^{\cdot}\mathrm{i}\iota\cdot 1\sigma- 111\mathrm{t})(]_{\langle^{\tau}}$$..\mathrm{i}_{\mathrm{l}1}]^{\neg}\mathrm{j}_{\mathrm{k})}.\triangleleft(.\iota)\rceil_{1_{\dot{\mathrm{f}}}1\mathrm{b}^{\backslash }}.(‘\}])\rceil(^{1},\mathrm{b}^{\backslash }\mathrm{S}11\mathrm{l}(’ 111’\mathrm{c}\mathrm{t}\mathrm{x}\mathrm{i}\iota 1111\mathrm{l}11\mathrm{i}_{\mathrm{l}\mathrm{I}}11\mathrm{I}\mathrm{t}^{i}111\mathrm{i}(1(11\mathrm{t}^{\backslash }$. $[2]$ .

$\mathrm{j}_{11}1^{}’ \mathrm{i}_{\mathrm{b})}.41))\mathrm{i},\mathrm{b}‘ \mathrm{s}\mathrm{i}\mathrm{l}11\mathrm{i}]_{i)\mathrm{l}}\mathrm{t}()\iota 1_{1\mathrm{t}^{\backslash }\mathrm{b}()}.|\mathrm{t}[1\mathrm{i}()|]\mathrm{i}_{\mathrm{l}1}]_{\mathrm{t}\mathrm{t}^{\mathrm{Y}}}^{)}\mathrm{I}^{\cdot}$.

$()\mathrm{f}\cdot \mathrm{t}]_{1\mathrm{t}^{\mathrm{Y}}}$$111()(1(^{1}1$$r\rfloor\urcorner]_{1(}\backslash$’”$]^{\urcorner}]_{1\mathrm{t}^{\backslash }\mathrm{f}1_{r}1\langle}$$\cdot\cdot]_{\mathrm{l}}\mathrm{i}_{1\mathrm{l}}\mathrm{g}- 111()(1(^{\tau}1^{\cdot}$

126
The ”flat model” in Fig. 4 c) corresponds to the solution in Ref. [16], where monodisperse rounded particles are ordered on a hexagonal grid. The angle of the arches canbe introduced as an additional parameter together with the angle of repose. The criticalangle can then be manipulated, so that vanishing pressure at the sides of the heaps is
obtained in Fig. ), ). Of course, such a phenomenological methodology allows the superposition of all three pressure distributions and the ”reproduction” of any experimental
outcome. Nevertheless, the models are only descriptive, they lack the power to predict theoutcome of a given experiment with a given material, considering that experiments withand without pressure minimum have been reported. Also models for the pressure distribution like the one given in [6], where one half of the system was fitted to a double-peakstructure, lack predictive power as to where and why a pressure dip occurs.
$4\mathrm{d}$$\mathrm{e}$Problem outline

1.4

A better understanding of the formation of the pressure minimum requires studies on
the micromechanic level. This includes the role of e.g. force network, friction, particle
shape and roughness, size dispersion, packing density etc. Up to now, neither from thesimulations nor from the experiments a complete consistent picture evolved as to what
constitutes the exact mechanism for the formation of the pressure minimum. Therefore,
we chose to investigate the problem with combined simulational and experimental methods to obtain insights beyond the current theoretical and experimental investigations.
Whereas it is now established thatthe the pressure minimum is causedby the wedge-sequence building history of the heap, it is not clear how
the information about this history is
stored. Possible mechanisms can be
imagined as loading of normal contacts between grains, the mobilization of static friction in the tangential
direction, packing densities, structure of the force network and so on.
, we
Before we address thesewant to clarify some points which arestill ambiguous in the existing exper, at
imentalleast on the micromechallic level.
lninimuIn is Figure 5: Pressure distribution for a simulation ofBecause theheaps, wc monodisperse round particles (lcss than 8wcll established for
dcviabstaincd from looking for thc ]) as ation in diamcter) under a
sct up in a layered
averagc of sequence.
a result of aotherwise noisy ta likc in Rcf. [17].
andin the ]singlc
The ])$\mathrm{q}\iota \mathrm{l}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$$\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}[5,6,7,8,9,10]$$\mathrm{p}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{s}\iota \mathrm{l}\mathrm{r}\mathrm{e}$$\mathrm{s}\mathrm{i}_{11}\mathrm{g}\mathrm{l}\mathrm{e}$$0\nearrow()$$\mathrm{d}\mathrm{i}$$\mathrm{h}\mathrm{e}‘ \mathrm{a}_{\mathrm{I}^{)}}$$\mathrm{c}\cdot \mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}11\mathrm{r}\mathrm{a}\mathrm{t}_{\backslash }\mathrm{i}()11$$\mathrm{c}\mathrm{l}\mathrm{a}$$\mathrm{d}\mathrm{i}$ill$\mathrm{s}\mathrm{l}_{1}on1\mathrm{d}\rceil_{)()}\mathrm{p}\mathrm{r}\mathrm{o}11\mathrm{O}111(.\mathrm{t}_{\mathrm{z}}^{1}\mathrm{c}1(^{\lambda}1^{r}(^{1}11\mathrm{f}\mathrm{t}\mathrm{J}\mathrm{l}\cdot\prime \mathrm{c}1$.

$11_{1(^{\backslash \mathrm{t}’}},,\mathrm{i}1\mathrm{r}111_{-}1\mathrm{c}\eta \mathrm{t}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{J}$$((11\mathrm{f}\mathrm{i}_{\mathrm{b}^{\eta\iota \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}_{\}1\mathrm{t}\mathrm{h}}})\langle)$$\mathrm{J}\mathrm{c}\mathrm{r}\mathrm{i}111(^{\mathrm{Y}}\mathrm{n}\mathrm{t}$$\mathrm{e}\mathrm{x}$127
Setup of the simulations and experiments

1.5

Granular cones and granular wedges built in a wedge sequence exhibit similar behavior [12].
Therefore, we considered that the computational treatment in two dimensions was sufficient to understand the microstructure of the heap.
To simplify the investigation, we assume that the bottom of the heap is neither deformed by external mechanism, neither by bending like in Ref. [18] or compression along
the -axis like in Ref. [17]. This is certainly the case for the setups used in [6, 8, 12]
and for our own experiments. For the simulation, we choose not to fix the outermost
particles, like in all the experiments we cited, in contrast to computational investigations
like Ref. [19].
If not otherwise noted, all heaps in the experiments and simulations are built in awedge-sequence, the grains are poured from a point source. All heaps are constructed
on a flat ground, i.e. a ground without any surface asperities. In the simulations, theYoung’s modulus of the ground is the same as for the grains and the friction coefficient
between ground and particles is the same as between the grains to simplify matters.
The setup of the experiments is described in appendix , the simulation method is described in appendix B. For all graphs, experiment and simulation, the ground pressures are
given in dimensionless units by rescaling the bottom pressure with l/(g*height*density).
Because the experimental pressure measurements used pressure gauges at different points,
we graphed the results using point symbols. The ”measurements” in the simulation havebeen made using ”moving averages” As these can be computed for any position of theline element, we drew the pressures with continuous lines.
$\mathrm{x}$$\mathrm{A}$Grain properties for pressure dips2

Because in the literature a variety of materials and setups has been used for the investigation of the pressure under heaps, ranging from rape seed over fertilizer, sand to glass
beads, we first wanted to investigate which grain properties are necessary requirements
for the formation of the pressure dip.

2.1

Monodisperse particles and la.yered sequence

For heaps of monodisperse smoothgrains in the absence of cohesion, no
pressure minimum in the middle is:.
observed. For this situation’crlation in Ref. [1C], gives constant. .:, which cleill thescribcs thc systcm quitc well. The
lcngth of the zone with constant pres- Figure
for for the sinlulation in Fig. 5. together with the outlinc of the hoap.
on thc anglc ofis )(( norlll‘d2
in )aodof tlle )jll ioll (. . If i,s )
in
.
to
, wc

. :

$:.\backslash \cdot$$\mathrm{c}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{l}-$$iil’ Jt’ t’ f”*’ \mathrm{t}iii^{ij}" \mathrm{f}^{\iota\iota\backslash \backslash \backslash }\mathrm{I}\mathrm{t}\mathrm{f}\mathrm{i}\mathrm{f}**\cdot.\backslash \backslash \backslash \cdot\backslash \prime i\prime\prime\backslash *\backslash \backslash \backslash \backslash \backslash |l\backslash \backslash \backslash \backslash \backslash \backslash \cdot...\backslash \cdot\cdot$$\mathrm{i}$,$\coprod 1\mathrm{i}\mathrm{d}\mathrm{d}1(^{\mathrm{Y}}$$\mathrm{p}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{l}\cdot \mathrm{e}$$C:\mathrm{S}\mathrm{t}_{}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{s}$$\mathrm{S}11\Gamma(^{\backslash }(1\mathrm{c}^{\mathrm{Y}}])(_{/}^{\mathrm{Y}}\mathrm{n}\mathrm{d}\mathrm{s}$$\mathrm{r}\mathrm{c}_{\mathrm{P}^{()\mathrm{S}\mathrm{C}}}$$|$$\ulcorner \mathrm{I}^{\urcorner}1\downarrow \mathrm{i}^{\mathrm{t}^{\tau}}‘$’$\mathrm{Q}\mathrm{c}\mathrm{a}1\iota \mathrm{s}(^{\backslash }1)\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathfrak{c}’ \mathrm{l}\mathrm{e}\mathrm{s}(1^{\cdot}\mathrm{d}\mathrm{c}\mathrm{l}\mathrm{O}11\mathfrak{c}1$$1\mathrm{r}\mathrm{i}_{\mathfrak{c}11\mathrm{l}}‘ \mathrm{g}n1_{\mathrm{d}1}‘\cdot 1_{\mathrm{c}}^{\Gamma})\{\mathrm{t}\mathrm{i}(\langle^{1}$$\mathrm{t}11(^{1}\mathrm{I})\mathrm{r}$’$\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}11^{(\backslash }‘\backslash$$\mathrm{l}(])\mathrm{a}\mathrm{g}\mathrm{a}\mathrm{f}\mathrm{t}^{\mathrm{Y}}(1‘ C11_{01\mathrm{l}}\mathrm{g}\mathrm{l}11$$\mathrm{s}‘ \mathrm{s}\iota 1Y\mathrm{t}^{1}$$‘ \mathrm{c}1\Gamma-$$\mathrm{I}$$\mathrm{t}\mathrm{i}\mathrm{c}\cdot 1\mathfrak{c}^{\backslash }\downarrow \mathrm{s}^{1}.1\mathrm{i}\mathrm{k}(^{\tau}$$\mathrm{I}$$\mathrm{T}\mathrm{l}1\mathfrak{c}^{1},\mathrm{s}^{\mathrm{t}}\mathrm{i}1\mathrm{l}11\iota 1$$l^{\sim}\urcorner \mathrm{i}\mathrm{g}.\cdot 4\mathrm{C}^{\backslash )}$$\Gamma$$\mathrm{s}^{\backslash }n1\uparrow,\mathrm{b}^{1}\mathrm{c}‘ 11\mathrm{t}^{\backslash }\mathrm{g}\mathrm{i}\backslash \mathit{1}\langle^{\backslash }11\mathrm{i}11\mathrm{I}\prec \mathrm{i}\urcorner \mathrm{g}$$5$$\mathrm{O}\mathrm{S}_{\backslash }\mathrm{s}^{1}\mathrm{i}]_{)}]‘\backslash$$1$$\iota\prime 1\rangle \mathrm{l}(^{\backslash }.\mathrm{b}^{\backslash }‘ \mathrm{b}[]\langle^{\backslash }(1\mathrm{i}_{])}\mathrm{i}\mathrm{l}\iota \mathrm{t}1_{1}\mathrm{i}_{\mathrm{h}^{\backslash }},\mathrm{f}\mathrm{i}\mathrm{g}\iota\iota \mathrm{l}\mathrm{t}^{\backslash }, |)\iota\iota \mathrm{t}.\dot{\mathrm{c}}1^{\mathrm{t}_{)}^{\mathrm{t}}}‘ \mathrm{t}11(^{\mathrm{t}}(1\mathrm{i}_{1)}\mathrm{i},\mathrm{b}^{\backslash }‘ \mathrm{b}’ 111’\mathrm{c}\backslash 11(^{\backslash }1^{\cdot}\mathrm{t}11\mathrm{a}\iota 111_{1(^{\tau}}\mathrm{f}\mathrm{l}\mathrm{t}\mathrm{l}\mathrm{c}\cdot \mathrm{t}\iota \mathrm{t}\mathfrak{c}‘ \mathrm{t}\mathrm{l}\mathrm{i}o11,\mathrm{b}^{1}$($\mathrm{i}_{\ln \mathrm{c}1}‘ \mathrm{g}\mathrm{i}_{11}\mathrm{e}$$(^{)}(11\mathrm{s}\mathrm{i}\mathfrak{c}1()\mathrm{r}$128
it not significant, in consideration of the experimental and computational results we will). The preference forshow later on. The histogram of the contact angles is given in Fig.
direction along
contact
preferredfrom
theordering on a hexagonal grid is clearly obvious
60-degree axes. For monodisperse rounded particles with friction on a smooth surface,
the difference in the pressure distribution to a system set up on an ordered
without friction on a fixed base are negligible. We can also compute the stresses inside theheap after Ref. [22], see Fig. 6 (for volumes of about $10\cross 10$ particles), but the results forstresses and the comparison with analytical theories will be given in a later publication.
$14\mathrm{b}$$\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}[20,21]$

2.2

Artifacts for smooth particles

129

130
2.4 Effect of cohesion andparticle roughness
From the previous section, we can aspressure dip for partisume$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}_{}\mathrm{t}_{1}\mathrm{h}\mathrm{e}$were not caused by some polydisper$00.\cdot.\cdot 20.30.40.500.700.05^{*}0607^{*}\ovalbox{\tt\small REJECT}^{*}18*90809\nabla^{**}\nabla\nabla\nabla\nabla$

glass bead with grain diameters be,i.e. for moderate rescaling (below t,hesystem size the physical properties
like pressure distribution and so on Figure 11: Pressure distribution for glass beads
will not change. Cohesion due to the and an average grain diameter of about 460
humidity in the surrounding air intro- diameter and large polydispersity $(210- 710 \mu \mathrm{m})$ .
duces a new length scale which vio- Two different lneasurements at the same position,
lates these scaling properties for par- the line is drawn through the average of the dots
ticles which are small small enough. to guide the eye.
In Ref. [11], where non-cohesive particles have been investigated via computer simulations, only polydisperse systems exhibited a pressure minimum.
In contrast, cohesion changes
movelnent of grains from single parti.

$\mu \mathrm{m}$$\mathrm{t}\mathrm{I}_{1}\mathrm{e}$$0.20.4\ovalbox{\tt\small REJECT}_{\mathrm{o}\mathrm{h}^{-}}0.60\mathrm{k}_{\mathrm{c}}-210^{4}\mathrm{N}/\mathrm{m}$

0.6
0.4
0.2

$\mathrm{k}_{\mathrm{c}\mathrm{o}\mathrm{h}}=4\cdot 10^{4}\mathrm{N}/\mathrm{m}$

$0$

0.6

$\mathrm{i}\mathrm{n}\mathrm{R}\mathrm{e}\mathrm{f}[23,24]\mathrm{A}\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{n}- \mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{w}\mathrm{a}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{b}\mathrm{y}\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{c}1\mathrm{e}\mathrm{s}\mathrm{t}_{}\mathrm{o}.\mathrm{c}1\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}.\mathrm{o}\mathrm{f}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}1\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}1\mathrm{e}\mathrm{s}$vestigation by computer simulat.ionsin Ref. $[25, 26]$ corroborated t,his picture. Fronu the non-uniformity of thecohesive force resnlts a disorder of thepacking and a polydisperse lnixt. Therefore, a system withofllonodispcrse cohesive particles beof
haves effectively like a)
12,. In Fig.
the indisperseof the pressure dip for increas(ohesionis shown. Nev-$\iota \mathrm{r}\mathrm{e}$$0.4$$\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{t}^{1}\mathrm{r}\mathrm{s}$0.2$\mathrm{k}_{\mathrm{c}\mathrm{o}\mathrm{h}}=12\cdot 10^{4}\mathrm{N}/\mathrm{m}$$\mathrm{k}_{\mathrm{c}\mathrm{o}\mathrm{h}}=l2\cdot

$0$

0.6

-

$0.4$

-

$\mathrm{s}_{\omega}\mathrm{v}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}1$$\mathrm{p}\mathrm{o}1\}^{I-}$$\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{s}$$\mathrm{p}_{\mathrm{c}}^{\mathrm{f}}$0.2$\mathrm{k}_{\mathrm{c}\mathrm{o}\mathrm{h}}arrow-20\cdot 10^{4}\mathrm{N}t\mathrm{m}$$\mathrm{c}\mathrm{r}\mathfrak{c}^{\mathrm{J}}\mathrm{a}\mathrm{s}\mathrm{e}$

0.4

0.2

$0_{0}$

1

0.8

0.6

$\mathrm{s}\mathrm{t}\mathrm{r}\in\backslash \mathrm{n}\mathrm{g}\mathrm{f}1_{1}$$\mathrm{i}\mathrm{l}\iota \mathrm{g}$$(^{1},\mathrm{r}\mathrm{t}1_{1}\mathrm{c}1(_{}^{1}\mathrm{S}_{\mathrm{k}}\mathrm{b}^{i}, 110\{\langle)\mathrm{n}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{h}()1)\mathrm{r}\mathrm{c}\mathrm{s}_{:}\mathrm{s}^{\mathrm{t}}\iota 1\mathrm{r}\mathrm{e}^{1}111\mathrm{i}_{11-}$

Figurc 12:$\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{s}_{\iota)}^{\mathrm{c}}’\iota 1\mathrm{r}\mathrm{e}\mathfrak{c}\mathrm{l}\mathrm{i}\mathrm{S}\mathrm{f}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{t}11\mathrm{c}\mathrm{l}\mathrm{c}^{1}\mathrm{r}\mathrm{h}\mathrm{e}‘ \mathrm{a}\mathrm{I}$)$\mathrm{s}$for imtlm,$[\mathrm{s}^{\backslash }\mathrm{i}]11111\mathrm{c}\gamma(\mathrm{i}\mathrm{t})11‘ \mathrm{b}^{\backslash }()\mathrm{f}\mathrm{n}1()\mathrm{n}\mathrm{o}(1\mathrm{i}\mathrm{s}1)\mathrm{c}\mathrm{r}_{\iota\backslash }^{\mathrm{c}\backslash }\mathrm{e}^{\mathrm{Y}}\mathrm{P}^{\mathrm{c}1\mathrm{r}\mathrm{t}\mathrm{i}\cdot]_{(_{t}^{\backslash \zeta^{\backslash }}}}‘(, (7\mathrm{c}()1^{\cdot}11\mathrm{t}^{\mathrm{Y}}\mathrm{r},\mathrm{b}.)$$1$$\mathrm{t}^{\mathrm{t}}Y\mathrm{t}^{1}\mathrm{c}‘ 1_{\iota\backslash }^{\mathrm{C}^{1}}\mathrm{t}^{\backslash },$)$\iota\iota \mathrm{t}\mathrm{a}\mathrm{J}\mathrm{s}\mathrm{o}$$:^{\backslash },()$

il is

the

$\mathrm{H}\iota \mathrm{l}\mathrm{c}\mathrm{f}\mathrm{t}\mathrm{l}’\mathrm{d}$tions ill-$(]_{\mathrm{C}^{\tau}}|)\mathrm{a}\mathrm{t}‘ c11)1‘\supset \mathrm{w}1_{1\langle^{\mathrm{Y}}}11\iota(^{1}1^{\cdot}\mathrm{f}1_{1(^{\mathrm{Y}}}$”$1\mathrm{t}^{\backslash }\mathrm{f}\cdot \mathrm{t}\mathrm{w}c‘ 11^{\cdot}\mathrm{d}()\mathrm{f}\cdot \mathrm{t}1_{1}\mathrm{e}^{1}1\mathrm{I}\mathrm{l}\mathrm{i}(1\mathrm{t}1]_{(^{\mathrm{Y}}}\langle)\{11_{1\mathrm{t}^{\backslash }}$$\mathrm{c}1r11(1(1\mathrm{i}\mathrm{f}\mathrm{f}(^{\backslash }\mathrm{I}()\mathrm{l}\mathrm{l}\{(.()11(^{1}h\mathrm{i}()\mathrm{l}1,\mathrm{C}_{\backslash }^{1(1(^{\backslash }11}.\mathrm{b}^{1\mathrm{t}]_{1}}\cdot \mathrm{T}\}_{1\mathrm{t}^{i}}\mathrm{I})\mathrm{l}’(^{1\mathrm{t}^{\backslash }\mathrm{C}_{)}^{\backslash }}|’‘\iota 1\mathrm{r}\mathrm{e}(1\mathrm{i}_{\rceil)}$$\mathfrak{c}1‘\backslash \backslash ^{\tau}(^{\backslash ]_{()}}\mathrm{I})\mathrm{S}\backslash \backslash ^{\mathrm{Y}}\prime \mathrm{i}\mathrm{t}1_{1}$$\mathrm{l}|()\iota\epsilon‘$

lt.l$\iota\iota \mathrm{a}1$

]

itlt

.

)

$1_{1\mathrm{t}_{(}^{1}\mathrm{t}\rfloor)}‘ \mathrm{i}_{1\mathfrak{l}}\mathrm{t}]_{1(}\backslash ]_{()\mathrm{W}(_{\downarrow \mathrm{h}^{1[}}^{\backslash }}\mathrm{b}$$1_{(}’.1$$‘ 1^{\cdot}\mathrm{t}^{1}\dot{\mathrm{c}}1\mathrm{s}^{1}\mathrm{i}_{11}\mathrm{b}^{\{()}’\cdot 11\mathrm{t}_{7}^{i\zeta^{\mathrm{t}}},\mathrm{i}()11$$\}^{r}\mathrm{d}’]_{\mathrm{c}11\not\in^{)})}.()\mathrm{f}\mathrm{l}\iota 1()(1\mathrm{l}\mathrm{r}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{t})11$.

$\mathrm{t}\mathrm{l}\mathrm{i}_{1)}.,$$1_{11},\mathrm{t}_{\backslash ()\mathrm{l}\mathrm{l}\downarrow\langle^{i}}^{1}\mathrm{C}\partial_{1}\mathrm{S}^{\mathrm{t}}\mathrm{t}_{\iota}^{1}\mathrm{b}^{\backslash }$.

])$1\iota()1^{\cdot}1^{\tau}\urcorner \mathrm{i}_{t}(),\cdot\rfloor^{\underline{)}}\mathrm{i}_{\mathrm{h}^{1}}$‘$\uparrow \mathrm{V}_{1}^{\mathrm{v}}()\iota\iota|\mathrm{i}_{\mathrm{I}1}\mathrm{i}_{11\mathfrak{i}\mathrm{c}1}‘$$\mathrm{i}_{\mathrm{t})\mathrm{j}}.\backslash ’(^{\backslash }1^{\cdot}\}^{r}|‘\backslash ^{1}1\iota\cdot\cdot()1(.()1_{\mathrm{t}(^{1}\mathrm{h}},’ \mathrm{i}()\mathrm{l}\iota,$131develop. The interpretation that cohesion affected the system for particles with 230
diameter is consistent with the fact that the bulk density for these small beads was, which was definitely smaller than the bulk density of the other glass beads(see Tab. 1). As the cohesion inhibits the compaction, therefore leading do smaller densities, could already be observed by previous.
The influence of the humidity of the surrounding air on granular materials has been
investigated already in Ref. [27], where the static friction has been observed to show atime dependence. Because it is nearly impossible to perform ”absolute dry experiments”
because of static electricity, we performed instead simulations of monodisperse particles
with cohesion. The algorithms are described in Ref. $[25, 26]$ . As can be seen in Fig. 12, forincreasing cohesion strength the dip increases, which can be attributed to the increased
effective polydispersity of the clusters made up from single particles.
In some experiments in powder
technology like in the Hosokawa powder tester (as proposed by Carr [28]),
the strength of the cohesion and thesurface roughness of the particles isclassified by a single parameter, theflowability. This flowability seems
to be responsible why a system with460
diameter glass beads and lessthan 8 % deviation (Fig. 10) in diameter behaves is a lnonodisperse syst,em and exhibits no pressure dip,
whereas sand with the same diameter and size dispersion shows a pressure dip (Fig. 13). For the same reason, the monodisperse frosted glass
beads in Ref.[8] with about 1 mm di- Figure 13: Pressure distribution for sand with 463
ameter dianueter showed a pressure, monodisperse $(425- 500 \mu\ln)$ .
averageminilnuln. Analogously, in a com- Two different realizations of the same measureputer simulation a system composed lnent, the line is drawn through the average of theof monodisperse rough or elongated dots to guide the eye.
particles can show a pressure dip, bereordering for rotlgh and elongated part.icles due to geometric const,raints is lnore
than forand round particles. Macroscopically this has been observed in
as a higher
colnp\iota lter
stress-for snlooth elliptic particles in colnparison
to snlooth round.

$\mu \mathrm{m}$$1.579\mathrm{g}/\mathrm{c}\mathrm{m}^{3}$$\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}[25,26]$$\mu \mathrm{m}$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}_{}\mathrm{e}\mathrm{r}$$\mu \mathrm{m}$$\mathrm{c}\mathrm{a}\iota \mathrm{l}\mathrm{s}\mathrm{e}$$\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\iota 1\mathrm{l}\mathrm{t}$$\mathrm{s}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}$$\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{l}\mathrm{t}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$$\}^{r\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{d}}$$\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\dot{\mathrm{c}}\mathrm{l}\mathrm{e}\mathrm{s}[29]$2.5Distribution of the contact angles

The distribntion of (olltact angles is(($\mathrm{I}\dashv_{\mathrm{O}\Gamma 11\mathrm{t}^{i}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}111\mathrm{O}11(\mathrm{d}\mathrm{i}\mathrm{s}])(^{1}\mathrm{r}\mathrm{s}\mathrm{e}])\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\cdot \mathrm{l}\mathrm{c}\mathrm{s}}^{\urcorner}$$\mathrm{o}\mathrm{l}\mathrm{l}1$

cont

1

ie 1)]($(^{\tau_{6}}\mathrm{h}^{\mathrm{t}}.\mathrm{b}’ 11(^{\mathrm{Y}}(f\mathrm{i}_{\iota}\mathrm{b}^{\backslash }11^{\cdot}\mathrm{i}\mathrm{I})\iota\iota \mathrm{t}\mathrm{i}\mathrm{o}\mathrm{t}1\backslash \mathrm{v}_{\dot{\mathrm{c}}}\downarrow t\mathrm{b}^{1}$$1(_{\backslash }^{1}\backslash (’ \mathrm{l}$il)ed. ir ,$\mathrm{a}(\{\mathrm{t}\mathrm{i}\mathrm{J}\iota \mathrm{b})]_{(^{\backslash }\mathrm{b}}‘$$\urcorner 0\mathrm{J}’ 111(\mathrm{l}(^{1}])$’$\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}1_{\mathrm{L}}\backslash$(influcnced
thetion in. In$\mathrm{b}\}^{r}$$1\mathrm{c}^{\mathrm{Y}}‘ \mathrm{s}‘ \mathrm{s}^{1}\mathrm{t}1_{1}\mathrm{a}112^{(/_{(}}\mathrm{d}\mathrm{e}\backslash ’ \mathrm{i}_{C}\gamma$$\mathrm{i}\iota 1‘ \mathrm{b}^{\backslash }(^{\backslash }(\{\mathrm{i}\mathrm{o}11\sim \mathrm{Q}.1$$\mathrm{s}^{\backslash }\mathrm{t}\mathrm{l}(\iota\iota \mathrm{g}1\backslash ’])(^{\backslash }‘ \mathrm{d}\mathrm{J}\mathrm{e}(^{\backslash }\mathrm{c}111(^{\backslash }‘ \mathrm{d}1^{\cdot}\mathrm{t}1\iota(^{i}\lambda^{r}\mathrm{c}\mathrm{I}$$\rangle[\backslash (1\mathrm{i}‘ \mathrm{b}^{\mathfrak{l}}])(^{\backslash }\mathrm{t}\mathrm{h}^{\tau}\mathrm{t}^{\backslash }])_{(11\mathrm{f}\mathrm{i}\mathrm{c}}’\cdot]_{(^{\backslash }\mathrm{b}^{1}},(\iota\iota])$to]$\mathrm{g}e\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\}^{r}$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{n}\mathrm{e}^{\mathrm{Y}}\mathrm{t}\mathrm{e}\mathrm{r}$$\mathrm{t}11\mathrm{i},\mathrm{h}^{\urcorner}\mathrm{c}\mathrm{a}\llcorner\langle^{)}\mathrm{t}_{)}^{\mathrm{t}}$),.

of thc particles.
)
effect$(_{\iota}\mathrm{s}^{\backslash }\mathrm{s}\mathrm{i}\mathrm{b}1()$$\mathrm{t}\mathrm{h}\mathrm{c}]$)$\uparrow 11\mathrm{C}^{1}1)\iota\cdot 01)\mathrm{a}1)\mathrm{i}\mathrm{l}\mathrm{i}\uparrow_{)^{r}}\mathrm{f}\mathrm{t}$.

$\iota\iota(^{)}‘ \mathrm{h}04^{\cdot}\mathrm{t}]_{1(}\backslash \mathrm{t}\mathrm{i}\cdot \mathrm{i}\mathrm{a}1n1\mathrm{e}11’ \mathrm{g}_{1}\cdot \mathrm{i}\mathfrak{c}1$$\underline{)}_{\lrcorner}^{\ulcorner}.(’/((1\mathfrak{c})\mathrm{Y}^{r}\mathrm{i}$($|1\mathrm{i}\mathrm{t}11\mathrm{i}\mathrm{l}\downarrow\langle\{\mathrm{i}\dot{\mathrm{r}}\dagger 1\mathrm{n}\mathrm{e}^{\tau}((^{1}\mathrm{l}).$tlll)$\mathrm{b}\mathrm{t}^{1}\mathrm{C}^{\backslash }$

(

$1$

)

Fig.

$\rfloor\angle]\mathrm{d}$$\mathrm{t}_{\dot{\zeta}}^{\backslash }\iota \mathrm{k}\downarrow \mathrm{s}^{1}\mathrm{f}()\iota$$1^{\cdot}$

).

. thc

132

(c) polydisperse, elongated(b) rounded particles, polydisperse (20 %)particles,(a)roundedmonodisperse

Figure 14: Histogram of the angle of the particle contacts versus the horizontal axis.
Graphs a) and c) are for wedge sequences, graph b) is for layered sequence.
triangular grid are still recognizable, but are already strongly smeared out, see Fig. 14 b).
For a size distribution of more than a factor of two, no structure is recognizable any morein the distribution of the contact angles, though the directions along the triangular grid
are still favored see Fig. 14 c). As a rule of thumb one can say that long as the angle
distribution is peaked around the angles of the hexagonal grid, in general no pressure dip
is found in the configurations.

3

Detailed Investigation of the heap structure

3.1

Force network

Force networks enjoy a certain popularity under theoretical physicist, due to their similarity to random walks, fractals and other geometric paradigm of theoretical physics.
Nevertheless, we were not able to extract any meaningful informations from force networks as given in Fig. 15, though the corresponding pressure distribution shows a rather
clear minimum.

3.2

Density of the heap

The experimental findings of the previous sections indicate that the pressurc lninimnIll isof the heap or fllea rather stable phenomenon. Perturbations like the )
)
(,
for
)
dip.
Even underof an inserted rod sccm only tohc )dip does not vanish tot ally. In
the ])
certain grain, of])vibrationa levelminimnm seems to be ingrained in.
, itorder of theof.
lllac ] (for a$\mathrm{r}\mathrm{c}^{\mathrm{Y}}1$$\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{l}\overline{\perp}\mathrm{t}$$\mathrm{u}\mathrm{i}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$$\mathrm{v}\mathrm{i}1$$\dagger$$\mathrm{r}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{c}\mathrm{p}\mathrm{r}$$\backslash r\mathrm{a}\mathrm{l}$$\mathrm{s}\mathrm{s}\iota 11^{\cdot}\mathrm{e}^{1}$$\mathrm{s}\mathrm{i}\mathrm{m}\iota 11\mathrm{c}‘’ \mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}^{\mathrm{C}^{\tau}},,,$$\mathrm{f}$$\mathrm{r}(^{1}\mathrm{s}^{\mathrm{C}^{}},,\mathrm{u}\mathrm{r}\xi^{\backslash }$$\mathrm{I}$$1^{\cdot}\epsilon^{1}\mathrm{s}\mathrm{s}\iota 1\mathrm{r}\mathrm{e}$$\mathrm{t}_{3^{\gamma}}\mathrm{p}\mathrm{e}\mathrm{s}$)$\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{l}^{-}\iota$$\mathrm{t}1_{1}\mathrm{c}$)$1\mathrm{i}\uparrow \mathrm{b}\mathrm{l}\mathrm{c}\mathrm{l}\mathrm{c}^{\zeta^{1}},$$\mathrm{f}_{}\mathrm{h}\mathrm{a}\mathrm{t}$$\mathrm{t}11(^{\mathrm{Y}}$$\mathrm{a}\ln$$\}_{1\mathrm{e}\mathrm{a}_{\mathrm{I}^{y\mathrm{t}11}}}$$(^{\lrcorner}‘ \mathrm{a}\mathrm{s}\mathrm{i}\mathrm{l}\gamma$$\mathrm{T}1_{1(^{\backslash }\Gamma(^{\mathrm{Y}}}\mathrm{f}_{01\mathrm{t}^{\tau}\mathrm{W}(^{\mathrm{J}}}]_{()()}\mathrm{k}(\backslash (1$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}-\mathrm{P}^{\mathrm{a}\mathrm{r}\{\mathrm{i}\cdot 1\mathrm{e}}(\mathrm{f})\mathrm{C}\mathrm{l}\mathrm{l}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{t})11\mathrm{t}\mathrm{l}\mathrm{e}^{1},]^{)}\mathrm{t}\mathrm{h}\mathrm{e}^{t}c11\ln()\{(1(_{\mathrm{t}}^{)\mathrm{C}},,\mathrm{t}\mathrm{r}\mathrm{o}.\backslash ^{r}$$r_{1^{\urcorner}\mathrm{i}\iota \mathrm{t}^{1}}(1_{i\mathrm{t}(_{(1}}()\{$$\mathrm{i}\mathrm{J}1^{\zeta^{\mathrm{t}}1(_{(}^{\backslash \prime})(1}‘$’$\cdot$$\mathrm{s}(()1)i((^{\backslash \mathrm{f}\mathrm{f},\mathrm{t}}\mathrm{t}^{\mathrm{Y}}(‘$$111\mathrm{O}1^{\cdot}\mathrm{C}${$(.()\mathrm{l}\mathrm{l}\mathrm{t}^{\Gamma}\zeta \mathrm{t}(.(11\mathrm{t}^{\tau}backslash 1^{\gamma}\mathrm{t})\mathrm{J}^{\cdot}\mathrm{k}])\Gamma \mathrm{t})\backslash r(\backslash$$\mathrm{l}11(^{\mathrm{t}}\iota‘ 111\mathrm{i}_{11\ltimes^{\mathrm{r}}}\mathrm{f}\iota 11$

et

$]$

fo

$\Pi 1\mathrm{i}\mathrm{t}1’ \mathrm{O},\mathrm{b}^{1}(()])\mathrm{i}((()111\mathrm{a}\mathrm{t}\mathrm{l}1()_{C}‘ \mathrm{J}_{\lrcorner}(1\mathrm{i}_{\mathrm{l}\mathrm{l}}\mathrm{g}\mathrm{c}^{1}\uparrow($.

$|_{)(^{1}}\mathrm{i}\mathrm{l}\mathrm{l}(.()1\mathrm{l}\mathrm{t}.1\iota \mathrm{l},\mathrm{h}.\mathrm{i}\iota^{\gamma}\mathrm{e}:\mathrm{D}_{11\mathrm{t})}1\mathrm{t})\mathrm{t}1_{1(^{)}11()}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{l}1(^{\tau_{\mathfrak{l}}}\mathrm{s},\mathrm{s}$$1\iota\iota(\mathrm{t}\iota\iota \mathrm{r}(^{\mathrm{Y}}(1h\mathrm{f}_{C}‘\iota\dagger \mathrm{i}^{\mathrm{t}},,,1\mathrm{i}(,\tau^{1}$.

$\backslash \backslash ^{7}(^{\backslash }\mathrm{v}’()|\iota \mathrm{l}\mathrm{r}\mathrm{l}\mathrm{l}\mathrm{l}()|(]_{(^{\backslash }}\iota^{}\mathrm{i}_{\mathrm{h}\mathrm{t}^{\mathrm{t}}r111)’}$

133

0.1

$0$

U.

0.2

$\mathrm{U}.\angle$$\rceil$

0.3

0.4

0.5

U.

U.4

U.4

$\mathrm{d}$X

X
Figure 15: Force network (middle) inside a heap with polydisperse particles built froma wedge sequence and the corresponding bottom pressure distribution (below). Stronger
forces are denoted by broader and lighter lines.
t,he heap was not homogeneous. By plotting the averaged
One possibility is
gavc an averaged pressllre dip (Fig. 16), we(Fig. 17) for heaps
density of thethe rest of the heap. This core
the middle region exhibits higher dcnsity
found
)
that,
middle.
minimum inshowed aclensity existed for allof higher
grainstheofimpactunderbulk dcnsity which fornls
core oftoItto the rcsulting avalanches which forms thethein tic wedge)
dip. Another interesting feature of the(.ore ofwllic ll results in)cllangcd
hcights is that tlle pressure
Fig. 16
in the, in (
tothewit
lninilntlm
anotin the. The )t,hc
of the
width
of
layer
(
higbeslis
rt fill,nearl.
$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}_{1}$$\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c},1_{1}$$\mathrm{h}\mathrm{e}\mathrm{a}_{1^{\mathrm{J}}}$$\mathrm{t}1_{1}\mathrm{a}\mathrm{n}$$\mathrm{t}\mathrm{h}c‘ \mathrm{i}\mathrm{t}$$\mathrm{t}_{\iota}\mathrm{h}\mathrm{e}$$\mathrm{b}_{11}1\mathrm{k}$$\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}$$1_{1}\mathrm{e}\mathrm{a}\mathrm{p}\mathrm{s}$$\mathrm{I}$$\mathrm{t}_{}\mathrm{h}\mathrm{e}$$1$$\mathrm{s}\mathrm{e}\mathrm{e}\mathrm{n}\mathrm{l};,\backslash$)$\mathrm{c}\mathrm{t}1_{1}\mathrm{i}\mathrm{s}$$1_{1}\mathrm{i}\mathrm{g}\mathrm{h}$$\mathrm{s}\mathrm{h}\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{i}_{1\mathrm{l}}\mathrm{g}\mathrm{c}\mathrm{l}\iota \mathrm{l}\mathrm{c}$$\mathrm{s}(^{\mathrm{Y}}(1^{1\mathrm{t}\mathrm{C}^{1}11\mathrm{C}\mathrm{C}}.\mathrm{d}11\mathrm{d}$””$\mathrm{t}1_{1\mathrm{C}}$$\mathrm{a}\mathrm{r}\mathrm{c}\cdot \mathrm{h}$$\cdot$$\mathrm{t}l1\mathrm{C}\mathrm{r}$$\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{s}\iota\iota 1^{\cdot}\mathrm{C}^{1}$$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{l}$$n\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{l}$$\mathrm{f}\mathrm{o}1^{\cdot}\backslash \gamma \mathrm{a}\mathrm{r}\mathrm{i}\mathrm{o}\iota 1_{\iota}\mathrm{b}^{1}$$\mathrm{I})\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}n\mathrm{r}\mathrm{c}\mathrm{d}\mathrm{i}1\mathrm{s}^{1}\mathrm{t}1^{\cdot}\mathrm{i}1)n\mathrm{t}\mathrm{i}\mathrm{o}\iota 1$$\iota_{1}\mathrm{t}11(^{\backslash }\mathrm{r}\in)1_{\mathrm{c}\mathrm{i}}^{\mathfrak{k}}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}]_{1(^{\tau}}\mathrm{i}\mathrm{g}1_{1}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{i}(1()$$\mathrm{h}\mathrm{e}\mathrm{a}_{\mathrm{I}^{)}}$$\mathrm{a}_{\iota}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{m}_{1^{)}}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}/\mathrm{p}\mathrm{r}\mathrm{c}\mathrm{d}\mathrm{i}(\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$$.()1\mathrm{l}\mathrm{t}\mathrm{r}\Re \mathrm{f}$$\mathrm{c}\mathrm{x}1_{1}\mathrm{i}\mathrm{b}\mathrm{i}\mathrm{t}$$1\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{t}111^{\cdot}\mathrm{t}^{\mathrm{Y}}[1,2,3]$$\mathrm{w}1_{1}\mathrm{i}(.\}_{1}$$1_{1}\mathrm{i}\mathrm{g}1_{1}\mathrm{c}^{1}\mathrm{s}\mathrm{t}1_{\dot{\mathrm{c}}1}\mathrm{y}\mathrm{c}\mathrm{r}(1\mathrm{o}(^{1}\mathrm{s}$$\mathrm{r}\mathrm{c}_{\mathrm{s}^{)}}^{\iota\backslash }\mathrm{s}\iota \mathrm{r}(^{1}$$\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s};\backslash ,\mathrm{u}\mathrm{r}(^{\mathrm{Y}}$$1$$.(11_{}‘,,\mathrm{i}\mathrm{s}\mathrm{l}\mathrm{t}^{\backslash }11\dagger \mathrm{w}\mathrm{i}\mathrm{f}1_{1}11_{1\mathrm{C}}\mathrm{f}_{\mathrm{d}(}‘\cdot \mathrm{t}\mathrm{t}1_{1}\mathrm{a}\mathrm{t}\mathrm{t}1\downarrow(^{1}$$\mathrm{Y}$$\mathrm{t}.1\iota\dot{\epsilon}\mathrm{i}111\downarrow(^{\backslash ]}()\mathit{1}^{\cdot}\mathrm{i}_{111])_{\mathrm{c}}’)(}\cdot 1\mathrm{i}_{1\mathrm{l}}\mathrm{g}])_{C}‘\}\mathrm{l}.\{\mathrm{i}_{\mathrm{t}}\cdot]_{(_{\llcorner}^{\backslash \mathrm{c}^{\gamma}}},\mathrm{i},\mathrm{s}’\dot{\mathrm{c}}1]" 1.\}^{(^{1}}r1^{\cdot}\mathrm{t}^{\mathrm{Y}}(1\mathrm{S}\mathrm{t}^{\mathrm{Y}}(1\mathrm{t}\mathrm{l}\mathrm{C}\mathrm{l}\mathrm{t}\mathrm{e}^{1}.‘\prime \mathrm{f}\mathrm{r}\mathrm{t})111\mathrm{t},1_{1(^{\mathrm{Y}}}\mathrm{p}\mathrm{o}\mathrm{i}_{11}\mathrm{t}()\mathrm{f}\mathrm{t}1_{1(^{\mathrm{Y}}}(.()11_{\Lambda}\mathrm{b}^{\backslash }\{1^{\cdot}\mathrm{u}\mathrm{t}.\mathrm{f}\mathrm{i}()11$$1\iota \mathrm{i}_{\mathrm{b}}|\iota_{()1\backslash }.,$

$1^{\cdot}\mathrm{c}‘\backslash \mathrm{t}1_{1\mathrm{t}^{\backslash }1}\cdot t]_{1\dot{\mathrm{c}}111}(‘|\backslash \backslash ’\langle^{1}(1_{\mathrm{b}^{)}\backslash }(^{\backslash }\mathrm{b}^{\backslash }\mathrm{t}^{\backslash }\mathrm{t}|11\mathrm{t}^{\backslash }1\mathrm{I}\mathrm{t}.\mathrm{t}^{\backslash }$.

$\backslash \backslash ^{\gamma}(^{1}1_{1\dot{\mathrm{c}}1\mathrm{t}}111\langle)\mathrm{i}_{\dot{\mathrm{r}}1(^{}}.\mathrm{i}\mathrm{l}\mathrm{i}1\backslash ^{\mathrm{v}}111\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{V}\mathrm{t}^{\backslash }\mathrm{t}\mathrm{s}’ \mathrm{t}\mathrm{i}\ell_{\neg})(.\}((^{\backslash }\mathrm{t}^{1}\mathrm{x}1)\mathrm{C}1^{\cdot}\mathrm{i}_{1\downarrow \mathrm{t}(^{\tau}11}|_{(}.\iota 11\backslash \cdot \mathrm{t}]_{1(})(1_{\mathrm{C}11^{\mathrm{c}\backslash }},,\mathrm{i}t.\backslash .’()[.

\mathfrak{j}]_{1\mathrm{t}^{\backslash }}]_{1\mathrm{t}^{\backslash }\dot{\mathrm{c}}1])\mathrm{h}^{\backslash }},.

it$)[. \mathfrak{c}1\prime 1)\mathrm{O}11\mathrm{t}8(/_{(}]_{1}\mathrm{i}_{\mathrm{t})}^{\zeta)}1_{1\mathrm{t}^{\backslash }1}\cdot(1(\backslash 11,\mathrm{s}^{\tau}$$(()\rfloor|1|)111,,\iota \mathrm{t}\mathrm{i}()1\mathrm{l}\dot{\mathrm{r}}11\mathrm{t}\mathrm{I}\mathrm{t}^{\mathrm{t}}\mathrm{t}1.\mathrm{b}\mathrm{i}\{\}’(\mathrm{t}\langle^{\backslash }\backslash \gamma \mathrm{i}_{i1}\mathrm{t}\mathrm{i}\mathrm{t})\mathrm{l}\mathrm{l}‘ \mathrm{s}\backslash \backslash ’ \mathrm{i}1]_{1}1\mathrm{i}’.\mathrm{f}_{1111\mathrm{J}\mathrm{l}\mathrm{t}^{i]}}..\langle$1)\iota\iota 1\mathfrak{j}]_{1\mathrm{C}}$$\iota’ \mathrm{i}\mathrm{l}\mathrm{l}\mathrm{t}1_{1\langle^{i}}$134$\mathrm{r}/\mathrm{d}$Figure 16: Pressure distribution averaged from 10 heaps with polydisperse particle size
distribution and wedge sequence. The pressure is measured at various heights and thecurve does not scale for pressures in higher regions.

0.76

0.79

0.78

0.77

Figure 17: Bulk$\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}_{}\mathrm{y}$

0.8

U.O1

$\mathrm{U}.82$

averaged from the 10 heaps of Fig. 16.

middle of the pile correspond reasonably well with the increased bulk density in thepacking distribution of a Catalyst bed filled from a point source in Ref. [30]. We did not
find any significant densit, differences for the heap build in a layered sequence, alld forno density differences weresuch a”rainy” uniform filling method, also
observed in Ref. [30].
$\mathrm{y}$$\mathrm{c}\mathrm{x}\mathrm{p}\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\downarrow \mathrm{a}\mathrm{l}1\}^{r}$

4

Conclusions

We have shown experimental alld computational evidcncc that for)is not homogcneous, )clip inthc(1 the)the (affect thea pressnreon thcof the)ofre$\mathrm{h}\mathrm{e}\mathrm{a}_{\mathrm{I}^{)\mathrm{S}\mathrm{W}\mathrm{i}\mathrm{f}1_{1}}}$$1$$\mathrm{t}1\downarrow\langle^{\mathrm{Y}}\mathrm{m}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{l}\mathrm{e}^{\tau}$$\mathrm{d}\mathrm{t}^{\backslash },\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t},\mathrm{y}$$\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{r}\mathrm{i}_{\mathrm{d}}]\iota\iota;,\backslash \mathrm{e}\mathrm{c}\mathrm{l},$$\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\cdot 1_{1}\mathrm{f}$$0\mathrm{l}\mathrm{h}$$\iota\iota \mathrm{t}\mathrm{d}\mathrm{e}\mathrm{I}$$(^{)}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{s}\mathrm{b}\mathrm{o}\mathrm{t}1_{1}$$\mathrm{W}\mathrm{i}\mathfrak{c}\mathrm{l}\mathrm{t}l1\mathrm{a}11(1$$\mathfrak{j}_{(^{1}])},\{]\mathrm{l}$$\mathrm{I}$$\mathrm{I})\mathrm{r}\mathrm{o}\mathrm{c}(^{\lambda}\mathrm{s}\llcorner \mathrm{s}^{\tau}\partial \mathrm{n}$$(1\mathrm{i}])$.

$\mathrm{h}\mathrm{f}\mathrm{i}\mathrm{e}\cdot 1’ \mathrm{t}$)$\mathrm{m}\mathrm{c}\mathrm{c}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{c}\cdot \mathrm{f}\mathrm{d}\mathrm{l}.\backslash \gamma$(,$m\mathrm{l}(1\mathrm{t}^{\iota}\mathrm{t}\cdot j\iota 11(^{\mathrm{Y}}\dot{\mathfrak{c}}1]).\mathrm{s}^{\mathrm{t}}$

is

$1$

)

$|^{\backslash }\mathrm{o}1\backslash \mathrm{t}^{i}\prime 11$.

$\zeta^{)}\mathrm{l}\mathrm{l}\mathrm{t}.\mathrm{s}$toAi )h’llowll.

.

$1$

)

$\mathrm{r}\{\prime 1\mathrm{t}^{\mathrm{Y}}‘,\mathrm{c}\mathrm{t}^{\backslash \zeta},$).

$\langle^{\backslash }$$\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{f}$$\mathrm{S}‘\partial_{-}\Gamma.\backslash .r\mathrm{r}(^{\backslash }\mathrm{t}]1\iota \mathrm{i}1^{\cdot}\mathrm{t}^{1}1\mathrm{n}\mathrm{e}^{1}\iota 1\mathrm{t}\mathrm{f}\mathrm{o}1^{\cdot}\mathrm{t}11()\{)((n\mathrm{r}\mathrm{r}\mathrm{c}^{\mathrm{Y}}11$

in the$\mathrm{c}\cdot 0\iota \mathrm{l}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{u}(\{\mathrm{i}\mathrm{o}\mathrm{n}1_{1}\mathrm{i}\llcorner \mathrm{s}^{}1\mathrm{o}\mathrm{r}\mathrm{v},$$\mathrm{b}\iota \mathrm{i}]_{(}1\mathrm{i}_{1\mathrm{l}}\mathrm{g}$($\mathrm{t}^{\mathrm{s}}$)$\mathrm{r}(^{\backslash },;\backslash ,\mathrm{s}\mathrm{t}1’ \mathrm{t}^{\backslash }$$1^{\cdot}\mathrm{C}^{\backslash }arrow\backslash ^{\mathrm{I}}\mathrm{s}n$$(1\mathrm{i}_{1})$$\mathrm{c}‘ 1\mathrm{I}$$\mathrm{t}1\downarrow(^{\mathrm{Y}}]_{1\mathrm{t}1110_{\mathrm{R}^{(^{i}11\{^{\backslash \mathrm{i}1\backslash }}}}, \mathrm{o}\mathrm{i}\cdot \mathrm{t}1\rfloor()])\mathrm{i}]_{(}\backslash$$1_{1}\mathrm{i},\backslash ((\mathrm{t}\mathrm{l}1(11)\langle^{1}\mathrm{a}c\mathrm{c}\cdot 0111\mathrm{p}\mathrm{l}\mathrm{i}^{\iota^{\backslash }},,1\iota \mathrm{c}\mathrm{d}()\mathrm{i}J\mathrm{t}]_{1(^{)}\mathrm{l}}\cdot|)\backslash \gamma])\mathrm{i}\mathrm{l}\mathrm{i}_{1\mathrm{l}}\mathrm{g}t]_{1\mathrm{C}}]1\mathrm{c}^{1}\mathrm{a}\})\mathrm{f}\mathrm{l}(111(‘\downarrow])()\mathrm{i}\iota 1\mathrm{t}$

135

136

Table 1: Materials used for the pressure measurements. The average diameter and the.
, the bulk density is measured indiameter are given in$[\mathrm{g}/\mathrm{c}\mathrm{m}^{3}]$$[\mu \mathrm{m}]$Acknowledgments
One of the present authors(H.-G. M.) is grateful for financial support of the Fellowshipsof the Japan Society for the Promotion of Science for Foreign Researchers in Japan. This
work is partially supported by the Grant-in-Aid for Scientific Research of Ministry of
Education, Science and Culture, Japan.

Appendix$\mathrm{A}$: Experiments

The heap was constructedusing a hopper with anoutlet of 6.2 mm which
was located 117 mm above
the ground plate, see
The experiFig. 19.
ments have been performed on stainless steel
ground plates of thickness 4.5 mm with pressuregauges of 12 mm diame- Figure 19: Experimental setup for the determination of the. different plates (355 ground pressure.
510 mm) have beenfrom the center, with a shift in
used where 4 or 5 holes were drilled in different, d5 were
the direction so to not interfere with each other. The distances
for plate 1 were $0,50,70,180,200$ mm, for plate 2 were 5, 30, 55, 165 mm, for plate 3
were 10, 40, 45, 120 mm, for plate 4 were 15, 35, 60, 85 lnm and for plate 5 were 20, 25,
100, 140 mm. The material properties of the grains are given in Tab. 1.
$\mathrm{t}\mathrm{e}\mathrm{r}$$5$$\cross$$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}_{}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{s}$$\mathrm{d}1,$

Appendix$\mathrm{B}$$\mathrm{d}2,$$\mathrm{d}3,$$\mathrm{d}4$: Simulation Method

contacts, wc used
To model and numerically silnulate cohesion as a feature of
dynamics. where the )crete element method. It is based onre. The ])
ion ofare colnputcd according to Newton’s
the
two )
forcc })
.
.
of
lhe
airea
alld to((io
is sket( ( in l$i\urcorner g. ).
.
as$\mathrm{t}11\mathrm{C}$$1\mathrm{n}\mathrm{o}\mathrm{l}\mathrm{e}^{1}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\mathrm{t}\}_{1\mathrm{e}}$(lis-$\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}(1\in)\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{j}\mathrm{c}\mathrm{e}\cdot \mathrm{t}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{e}^{\mathrm{Y}}\mathrm{s}$$\mathrm{I}$$\mathrm{m}\mathrm{o}\mathrm{t}_{}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}(\mathrm{l}\mathrm{e}\mathrm{S}’\mathrm{d}$$\mathrm{r}(^{1}])\mathrm{r}\mathrm{c}\mathrm{s}^{\backslash }\mathrm{t}^{\backslash }\mathrm{n}\mathrm{t}\mathrm{e}(1$$\mathrm{e}\mathrm{q}\iota \mathrm{l}_{(}’\iota \mathrm{t}$by,$\mathrm{Y}\langle)n1l\mathrm{s}^{\mathrm{t}}\mathrm{l}\mathrm{l}\urcorner()(1\iota\iota 1_{11_{\iota}^{\mathrm{c}}’}$$\mathrm{e}\mathrm{t},\mathrm{w}\mathrm{t}^{-\backslash }\mathrm{C}11$$\ulcorner \mathrm{I}^{\gamma}]_{1\mathrm{C}}$$\mathrm{p}o1\}^{\gamma}\mathrm{g}\mathrm{o}11\mathrm{S}$$1$$\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{e}^{\backslash }1()\mathrm{s}\{i, .j\}\mathrm{i}_{\mathrm{h}^{\backslash }},\iota)1^{\cdot}0])(\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{l}\mathrm{t}\langle)$$\mathrm{c}1(^{1}\mathrm{f}\mathrm{o}\Gamma 111r‘ \mathrm{t}\mathrm{l}\mathrm{i}\mathrm{o}\mathrm{l}|)$$\mathrm{t}1\mathrm{l}(^{\backslash }()\mathrm{v}\mathrm{e}^{\mathrm{Y}}\mathrm{r}\mathrm{l}\mathrm{a}])$$\mathrm{h}(^{\mathrm{Y}}$$\mathrm{T}1\mathrm{l}\{^{\backslash }$$\{\backslash \backslash ^{r}0_{1^{)\mathrm{o}1})\mathrm{g}_{011\mathrm{b}}}’‘$$(\mathrm{r}\mathrm{C}^{\backslash }])\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}^{1},111\mathrm{i}_{1\mathrm{l}}\mathrm{g}$$1$$2\mathrm{t}$) $111\mathrm{i}\dot{\mathrm{c}}11(1\mathrm{i}\iota \mathrm{t}^{1}\langle‘(\mathrm{i}()11$$r_{\mathrm{I}^{1}1_{1\mathrm{t}^{\backslash }}\mathrm{i}_{1}\cdot \mathrm{i}(\mathrm{t}_{)}\mathrm{i}(\mathrm{J}11}‘ \mathrm{c}\mathrm{t}$$.\mathrm{b}$$1_{\mathrm{c}1\mathrm{l}1}^{\mathrm{r}}\mathrm{g}\langle$$(.\langle)\mathrm{l}\mathrm{l}1_{\dot{\mathrm{c}}\{(}\mathrm{t}\mathrm{g}\mathrm{t}^{\mathrm{Y}}\mathrm{O}\mathrm{I}\mathrm{I}1(^{1}\mathrm{t}\mathrm{r}\backslash \gamma$’$rl,\mathrm{w}\mathrm{h}arrow,\mathrm{i}_{\mathrm{t}}\cdot 1_{1}\mathrm{i}_{\mathrm{S}(}\cdot]_{1()\mathrm{b}\mathrm{t}^{1}11},$$\mathrm{t}11\mathrm{t}^{\backslash }(1\mathrm{j}\mathrm{l}.\mathrm{t}^{\backslash }\mathfrak{c}\cdot \mathrm{t}\mathrm{i}()\mathrm{l}1()\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{t}^{\backslash }\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{f}(^{1}1^{\cdot}|\mathrm{S}(^{)}(\mathrm{t}\mathrm{i}()11])()\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{t}\mathrm{s}l_{1}^{)}.

”\Gamma_{2^{()}})‘ \mathrm{f}\mathrm{f}1_{1\langle^{1}}\mathrm{I}^{)()}1\backslash ^{\Gamma}\mathrm{k}^{\langle)||\backslash ^{\gamma}}$$\cdot\prime 1^{\tau}1_{1}\mathrm{i}\downarrow \mathrm{s}^{\tau}$

137, along which normal forces are acting. The force point fromdefines the directionof thewhich the contact forces act on the particles is chosen as the center of mass
overlap polygon. The velocity component resulting from the rotation of the particles. Additionally, damping in normal
is computed with respect to this ”contact point”
direction and friction in tangential direction is implemented.
More details of the force law for the contactand damping forces are explained in [11]. The
simulation was performed for a two-dimensional
system. We assumed that the cohesion strengthis proportional to the contact area. Then, for
two dimensions, the force is proportional to thecontact length (dotted line in Fig. 21) whereas
proportional
the repulsive force wasto the area overlap of two particles (shaded region in Fig. 21). For penetration spheres, thearea overlap and penetration depth are pro, so that the Hertz contact
portional as
for the contact force $F$ is recovered.
law
Figure 20: Sketch of the contact situa- Likewise, for penetrating angles, the continuum
tion for our polygonal collision law.
mechanically correct contact law withis reproduced. No rolling friction was implemented, because no coefficients are available in the literature for granular materials, andin the presence of corners the normal dissipations dampens away rotations anyway.
in two dimensions iswith units
For the cohesion, the cohesion strengthis proportional to the contact length :used, so that the attractive force$\vec{n}_{n}$$S_{ij}$$\omega_{i},$$\omega_{j}$$S_{ij}$$\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{n}[11]$

$A$

$x$

$A\propto x^{3/2}$$F\propto x^{3/2}$$F\propto x^{2}$$[\mathrm{N}/\mathrm{m}]$$k_{\mathrm{c}\mathrm{o}\mathrm{h}}$$l$$F_{\mathrm{c}\mathrm{o}\mathrm{h}}$$F_{\mathrm{c}\mathrm{o}\mathrm{h}}=k_{\mathrm{c}\mathrm{o}\mathrm{h}}\cdot|l|$,(1)

In contrast to the model used in

138
In the simulation, the static and dynamic friction coefficients were chosen as.
$=0.6$
forradiuswas
0.00125,
particle,
Young’swasthemodulus.
, the density was 5000
the particles. The time step for the simulations was$\mu_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$$\mathrm{Y}=10^{7}\mathrm{N}/\mathrm{m}$$\mu_{dyn}$$=$$\mathrm{m}$$\mathrm{d}\mathrm{t}=0.2\cdot 10^{-5}\mathrm{s}$.
All simulations used a flat ground with friction, but without any surface asperities.
The pressures are computed using moving averages by computing the forces under line
elements of 15 to 30 particle diameters of length. The effects were rather stable with, the employed parameters and modified force laws.
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