HAL Id: jpa-00247031
https://hal.archives-ouvertes.fr/jpa-00247031
Submitted on 1 Jan 1994
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Computer simulations of granular materials: the effects of mesoscopic forces
G. Kohring
To cite this version:
G. Kohring. Computer simulations of granular materials: the effects of mesoscopic forces. Journal de Physique I, EDP Sciences, 1994, 4 (12), pp.1779-1782. �10.1051/jp1:1994115�. �jpa-00247031�
Classification
Physics Abstracts
02.70 46.10 62.20
Short Communication
Computer simulations of granular materials: the elilects of
mesoscopic forces
G-A- Kohring
Central Institute for Applied Mathematics, Research Center Jülich (KFA), D-52425 Jülich, Germany
(Received 12 October 1994, received in final form 13 October 1994, accepted 20 October 1994)
Abstract. Trie problem of trie relatively small angles of repose reported by computer
sim-
ulations of granular materials
is discussed. It is shown that this problem can be partially
understood as resulting from mesoscopic forces which are commonly neglected in trie simula- tions. After including mesoscopic forces, characterized by trie easily measurable surface energy, 2D computer simulations indicate that trie angle of repose should mcrease as trie size of trie granular grains decreases, an elfect not seen without mesoscopic forces. Trie exact magnitude
of this elfect depends upon trie value of trie surface energy and trie coordination number of trie
granular pile.
Trie study of granular materials bas long been an active field of research, partly due to the many interesting physical phenomena which granular materials give rise to and partly
because of their importance for industrial applications iii. Due to the advent of more powerful computers, especially powerful workstations, many scientist and engineers believe that some of the phenomena known in this field can be better understood through well planned computer
simulations [2]. This behef rests on the premise that these phenomena are collective or emergent in nature, i e., the constituent grains experience simple, well understood interactions with each
other, but that unexpected behavior emerges due to the large numbers of grains involved.
Hence, if the grain-grain interactions can be efliciently prograrnmed so that a sufliciently large system can be simulated, then it should be possible to study phenomena which are still poorly
understood.
The realization of this scheme then rests upon the use of the "correct" interactions in the computer simulations. Where "correct" may depend upon the types of answers in which
one is interested. Unfortunately there is at present no concept for granular matenals which
corresponds to the universality concept in statistical physics [3]. For this reason there is no
concensus as to how the grain-grain interactions should be modeled and how detailed this
modeling should be [2]. The situation is compounded by the fact that most simulations are still confined by a lack of computer power to two dimensions, where quantitative comparison
© Les Editions de Physique 1994
1780 JOURNAL DE PHYSIQUE I N°12
with experiment is not possible.
Computer simulations of granular materials using the molecular dynamics [4] (or "distinct element" [2] approachj generally assume that only short range, elastic interactions are present.
In what we will call the Hertz contact model for spherical grains [5], the repulsive force acting normal to the surface of two colliding grains is given by:
~~ ~a2 ÎÎÎ2 ~~ ~~ ~~'~~~~'
~~~
where Y is the Young modulus, a is the Poisson ratio, h
= d Ri R2j with d being the distance between the centers of mass and R~ the radius of the i-th particle, n~ is a unit vector normal to the surface of the particlesj mr = mim2/(mi + m2)j ~~ is the energy dissipation
rate and v£, is the relative velocity normal to the contact surface. o varies between 3/2 for smooth particles and 1 for rough particles.
For the forces acting tangential to the contact surfacej a standard approach is to assume that the Poisson hypothesis is true and to write:
Fll
= min (mr~ll v)'~~ ii ~ F~
[) r~l~~~ (3)
where ~" is the shear dissipation rate, v)l~, is the relative velocity tangential to the contact
surfacej and ~ is the friction coefficient.
The model based upon equations (1-3) does not show any qualitative changes with variations in particle size. However, as the particles become smaller and smaller~ atomic and molecular
forces will become important. But at what length scale will such forces begin to make a noticeable contribution to the physics of granular materials?
When the molecules of two grains interact via the van der Waals potential~ the resulting mesoscopic interaction is called the Hamaker potential and has been experimentally verified [fil.
However~ for most materials, there are other short-range atomic forces which are far stronger than the van der Waals forces. These forces can be characterized on the mesoscopic scale by
the surface energy~ W~ of the contacting surfaces and are attractive in nature. For the case of
general surface forces, Johnson et ai. derived the following expression for the contact of two spherical grains [7]:
~
31 a2 ~~~~
~
~~~
3 1 a2
~~~)
Ri + R2 ~~~
Obviously, equation (4) represents a force which is attractive for small values of h and repulsive
for larger values and indeed this was verified in subsequent experiments [7].
The maximum value of the attractive force in equation (4) is given by:
Fmax = (~rW/~~( (5)
1 +
2
For a typical value for W of W
r- 11~l~ erg/cm~, we can ask when this attractive force is equal to the weight of a particle. This occurs when 4/3~rR(g
= 3/2~rWRiR2/(Ri + R2). For particles with a density of 2 g/cm~ we find that the attractive force is equal to the particle weight when Ri is approximately: Ri * 2.5 mm. For grain sizes smaller than this value the mesoscopic
forces will make significant contributions to emergent behavior of the granular material. This
value of Ri may seem qmte large~ but is related to the initial assumptions: smooth, clean particles. If the particles are not clean W is reduced and if the particles are not smooth then
o is closer to one than 3/2. Hence~ for most real systems this critical radius would be reduced.
Now, one of the most popular experimental systems consist of glass beads having a radius of l~.15 mm [8~ 9]. These glass beads are an order of magnitude smaller than the critical radius calculated above. Hence, computer simulations which rely on equation il) alone may be inadequate for modehng these experiments.
One possibility for detecting the presence of these mesoscopic forces is through the so-called
angle-of-repose- Experimentalist and engineers have long used this quantity to characterize
granular materials, yet simulations have repeatedly found angles which are much smaller than the experimental values [ll~].
We have carried out computer simulations using equation (4)~ to determine the angle of
repose for a system of spherical particles. The simulations consist of dropping the particles one
at a time onto a flat plate and measuring the maximum angle obtained by the resulting pile.
The particles are log-normally distributed with a mean diameter of l~.15 mm and a standard deviation of l~.1~5 mm (The distribution is cut off at l~.21~ mm and l~.ll~ mm.) The Young modulus was taken to be Y
= ll~~ Pa and a time step of1.l~ x 11~~6 seconds was used.
Figure la shows the results for zero surface energy~ W
= l~ and those in 16 for W
= 31~
erg/cm~. IA relatively small value of W is used to mimic experiments where the particles may
° @fi
~ l dl ,,~
~ '' <( '» '~
t ~ .) l ~
~i< t Î ', .t '~'
o' f 4 < '~ à.
'&' ~~ '~ '~' ~.' 'ç ~~,
~>
t" S' " ', ,'<
= 'w'
<., ,j ~ _' ,i'.
~' ~,
MM, ~'< '.
~ÎÎÎ~
~Ù
a)
b)
C)
Fig. l a) Mean particle radius 015 mm. Plate is 15 mm wide Surface energy, W
= 0. Trie
angle of repose is approximately 14°. b) Same as la but W
= 30 erg/cm~. Trie angle of respose is
approxirnately 20°. c) Sarne as lb~ but mean particle radius is 15 mm and trie plate is 1500 mm wide.
The angle of respose is approximately 14°.
1782 JOURNAL DE PHYSIQUE I N°12
not be very clean.) Both systems were simulated for 121~ simulation seconds i-e-, 1.2 x 11~~ time steps. Each figure shows the largest slope obtained during these 121~ seconds. The slope in
figure 16 is approximately 21~° while that of figure la approximately 14°. Hence, the presense of mesoscopic forces leads to larger~ more realistic angles-of-repose- On the other hand, figure
lc shows a system with the same surface energy as 16, but where all particles are ll~l~ times
larger, 1-e-, the mean diameter is là mm. Here again the angle of repose is smaller than in 16 and nearly the same as in la, 1-e-, the mesoscopic forces are unimportant for larger particles.
Experiments indicated angles of repose on the order of 31~° for systems of glass beads with
the sizes given above [8]. However, m 3 dimensions~ the coordination number for each particle
Ii.e. the number of neighbors) is about SI~% larger than in 2 dimensions, so that particles on
the surface would feel a larger attractive force due to the mesoscopic part of equation (4)~ thus
leading to an increase in the measured angle of repose.
In conclusion, simulations show that the angle of repose should mcrease when the grain size
decreases, for clean~ smooth particles. As for as we know, this is the first time that such a
dependence has been explicitly postulated. All previous suggestions aimed at increasing the angle of repose found in computer simulations were made to work independently of trie grain size and independently of mesoscopic forces [11~]. It should be relatively simple for experimentalists
to test the validity of the present conclusions by measuring the angle of repose as a function of the grain sizes for materials with diffenng surface energies.
References
Ill Schwedes J
j
"Fheflverhalten von Schüttgütern m Bunkern" (Verlag Chemie GmbHj Weinheimj 1968)j
Rietema K-j "The Dynamics of Fine Powders" (Elsevier, London York, 1991);
Mehta A.j "Granular Matter" (Springerj Berlinj 1994).
[2] Cundall P-A- and Strack O.D.L
j
Geotech. 29 (1979) 47;
Walton O R and Braun R L.j J. Rheology 30 (1986) 949;
Thompson P-A- and Grest G-S
j
Phys Reu. Lent. 67 (1991) 1751.
[3] "Applications of trie Monte Carlo Method in Statistical Physics"j K Binder Ed. (Spnngerj Berlin, 1994).
[4] Allen M P. and Tildesley D.J.j "Computer Simulations of Liquids" (Clarendon Press, Oxford, 1987)j
Hockney R-W- and Eastwood J-W-
j
"Computer Simulation Using Particles" (Adam Hilgerj Bris- tol, 1988).
[5] Landau L D. and Lifschitz E-M
j
"Theoretical Physics"j Vol. VII (Akademiej Berhnj 1989) [6] Hamaker H-C
,
Physica IV1o (1937) 1058j Dahneke B-j J. Golloid Interface Sa. 40 (1972) 1 [7] Johnson K-L.j Bot. J. Appt Phys. 9 (1958) 199;
Johnson K-L-, Kendall K. and Roberts A.D,j Froc. R. Soc. Land, A. 324 (1971) 301.
[8] Rajchenbach J-j Phys Reu Lett. 65 (1990) 2221.
[9] Zik O., Levine D., Lipson S,G,j Shtrikman S, and Stavans J-j Phys, Reu. Lent. 73 (1994) 644.
[10] Lee J and Herrmann H-J-, J Phys. A26 (1993) 373,
Pôschel T, and Buchholt2 V.j Phys, Reu. Lent 71 (1993) 3963.
