EUROPHYSICS LETTERS 1 January 1999

Europhys. Lett., 45 (1), pp. 76-82 (1999)

Arching in confined dry granular materials

R. Peralta-Fabi1(∗), C. Malaga

1 and R. Rechtman2(∗∗)

1 Departamento de Fısica, Facultad de Ciencias, UNAM - 04510 Mexico, D. F., Mexico2 Centro de Investigacion en Energıa, UNAM - 62580 Temixco, Mor., Mexico

(received 7 July 1998; accepted in final form 28 October 1998)

PACS. 83.70Fn – Granular solids.PACS. 46.10+z – Mechanics of discrete systems.PACS. 05.40+j – Fluctuation phenomena, random processes, and Brownian motion.

Abstract. – We discuss a simple model that applies to a random array of arbitrarily shapedgrains, contained between rigid vertical walls, that predicts arching. That is, the pressure (orweight) of the column of grains saturates as its height increases. The average behavior of themodel is solved through a discrete and a continuum analysis. We find a qualitative agreementwith Janssen’s phenomenological result for arching. The saturating pressure grows with N2,where N is the horizontal size of the system. Adjusting our numerical results to Janssen’smodel we find a relaxation depth that also grows with N2. The results of the average behaviorallow us to measure fluctuations; the relative fluctuation of the pressure goes to zero as N−1/2.The continuum analysis shows that the weight inside the column satisfies a diffusion equationwith a source term and particular boundary conditions which leads to a complete solution. Thefirst-order approximation is similar to Janssen’s result.

Introduction. – Granular materials are present almost everywhere; sugar grains, a sandpile, seeds in a hopper, and food grains in a silo are just a few representative examples.Recent research has focused on different aspects such as describing avalanches [1], mixingand segregation [2], surface patterns in vibrated containers [3], and stress distributions [4-6],among many others. The central issue is building a well-established phenomenology and thecorresponding theory, at the small and large scales, aside from the practical importance ofunderstanding and predicting their behavior [6].

Depending on the external forcing, a granular system will behave as a solid, as a fluid,or in a way that seems to lack any resemblance to other many-body systems. Suffices tomention that a general phenomenological theory is still lacking, though various approacheshave been proposed to establish a general theoretical framework. Not surprisingly, besides thephenomenological approach to build constitutive relations using solely macroscopic concepts [7]or a mixed viewpoint [8], the conveniently adapted prescriptions of kinetic theory [9] andstatistical mechanics [10] have also been considered.

(∗) E-mail: peral@servidor.unam.mx(∗∗) On leave from Departamento de Fısica, Facultad de Ciencias, UNAM, Mexico.

E-mail: rrs@mazatl.unam.mx

c© EDP Sciences

r. peralta-fabi et al.: arching in confined granular materials 77

In this letter we address the problem of arching in dry granular materials. Arching refersto the observation that the pressure at the bottom of a column of grains saturates at a valuePs as the height of the column is increased. In other words, the normal (vertical) stressbecomes practically independent of depth [7]. Through a simple argument, Janssen [7, 11]showed that P = Ps(1− exp[−t/tr]), where t is the depth of the column, P the pressure andtr a relaxation depth. The model we discuss resembles the one proposed by Harr [12] andlater, independently, by Liu et al. [4], in which attention is directed to the force distribution.Recently, similar models that include both normal and tangential force equilibrium have alsobeen studied [13].

The weight of a column of grains depends sensitively on the way the column is built and thedifferences between experiments may be large. Indeed, slight changes in temperature have animportant influence on the weight [14]. The lack of information on the detailed way in whichstresses are propagated vertically is modeled using uniformly distributed random variables.As in statistical mechanics, we deal mostly with the average behavior which allows a completediscrete solution and in the continuum limit leads to an inhomogeneous diffusion equation thatcan be solved. The average behavior is easily obtained by substituting the random variables bytheir averages; this solution agrees to first order with Janssen’s result. It has been argued thatfluctuations in these type of systems are large. However, a numerical investigation showed thatthe relative fluctuations of the pressure around its mean value scale approximately as N−1/2,where N is the horizontal size of the system.

The model. – We propose a simple model of grains inside a rigid container, under theinfluence of gravity, and in contact with the walls. The model is a two-dimensional granularsystem in which each horizontal layer has N beads (disks) of unit diameter. In the first layer,starting from the top, the leftmost bead rests on both the left wall of the container and on thebead below it; the second bead rests on the two below it and so on. Correspondingly, in thesecond layer, the rightmost bead rests on the right wall of the container and on a bead belowit, the one next to it rests on the two beads below it and so on. This pattern is repeated foreven and odd layers of beads. In each layer, only one bead is in contact with one of the walls.

A way to symmetrize the analysis is to consider a staggered lattice; that is, two independentinterpenetrating arrays that form a perfect square lattice of width 2N . In this lattice, bothbeads at the ends are in contact with a wall, and even (odd) numbered beads rest on the odd(even) numbered beads from the next layer. There are no lateral forces between beads in thesame layer.

Each layer is labeled by t = 0, 1, ...; the sites in a layer are labeled by i = 1, 2, ..., 2N .The diameter of the beads, their weight and the distance between layers are all unitary anddimensionless. Each grain sits on two grains and partially supports two from the layer above.Disorder and variations in shape and orientation, as appear in real systems, are introduced as

follows. A site i at depth t+ 1 has a weight w(t+1)i , due to its own weight and the weights of

the corresponding (two) grains at depth t, according to

w(t+1)i = 1 + r

(t)i−1w

(t)i−1 +

[1− r(t)

i+1

]w

(t)i+1, (1)

with w(0)i = 1 and r

(t)i a uniform random number between zero and one. A grain at site (i, t)

shares a fraction r(t)i of its weight to the one at site (i + 1, t + 1) and the remaining fraction[

1− r(t)i

]to the one at site (i− 1, t+ 1). The boundary conditions are

w(t+1)1 = 1 +

[1− r(t)

2

]w

(t)2 , w

(t+1)2N = 1 + r

(t)2N−1w

(t)2N−1. (2)

78 EUROPHYSICS LETTERS

1e+03

1e+04

1e+05

1e+06

1e+07

1e+01 1e+02 1e+03 1e+04

tr

N

Ps

3

3

3

3

3

3

+

+

+

+

+

+

Fig. 1. – We show the average over 20 numerical experiments of the saturating pressure Ps (3) andthe relaxation depth tr (+) for N = 50, 100, 200, 400, 800, and 1600. We found that Ps = 0.679N1.998

(continuous line) and tr = 0.716N2.020 (dotted line).

These conditions model the role played by the walls, as they take away a fraction of the weightfrom the grains leaning on them. The pressure P is defined by

P (t,N) =1

2N

2N∑i=1

w(t)i . (3)

The goals are then to show that the model predicts arching, that is, to prove that Ps =limt→∞ P (t,N) exists, to determine how this limit is reached, its value, and to find therelaxation depth tr. The system satisfies mechanical equilibrium since no tangential stresses arepresent and there is no friction between grains; yet the boundary conditions can be interpretedas the frictional contacts with the walls.

Numerics. – By adjusting the numerical experiments with Janssen’s result we found thatboth the saturating pressure Ps and the relaxation depth tr scale with N2 as we show in fig. 1.This suggests that 〈P 〉/N2 is a universal function of t/N2, where 〈P 〉 is the average pressure,as we show in fig. 2 for several values of N . We also found that the relative fluctuation of thepressure σP (t) = 〈(〈P 〉−P )2〉/〈P 〉 goes to zero as N−0.493 ∼ N−1/2 for t� tr. The numericalexperiments show that arching is always present.

Discrete analysis. – Taking r(t)i = 1/2 in eqs. (1) and (2) is equivalent to averaging over

many realizations of the experiments, an ensemble average, as can be seen readily by takingaverages on both sides of these equations or by repeating the numerical experiments. In this

r. peralta-fabi et al.: arching in confined granular materials 79

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10

〈P 〉

N2

t/N2

Fig. 2. – We show 〈P 〉/N2 vs. t/N2 for N = 50, 100, 200, 400, 800, 1600 and 20 numerical experiments.The curve with the largest fluctuations corresponds to N = 50. The quantity 〈P 〉/N2 is a universalfunction of t/N2.

case, the pressure may be written as

NP (t,N) =N∑k=1

t∑m=0

a(k)m

2m, (4)

where the coefficients satisfy the recurrence relations a(k)m = a

(k−1)m−1 + a

(k+1)m−1 , a

(1)m = a

(2)m−1,

a(N)m = a

(N)m−1 +a

(N−1)m−1 , with a

(k)0 = 1. Without going through the details, the solution to these

equations can be expressed as

2−ma(k)m =

N∑i=1

Aisin(k + 1)θi

sin θicosm θi,

where θi = π(2i− 1)/(2N + 1), and the A’s satisfy

N∑i=1

Aisin(k + 1)θi

sin θi= 1; k = 1, ...N.

By summing over m in eq. (4), the pressure can be written as the difference between twoterms:

P (t,N) = Ps(N)− PN (t), (5)

80 EUROPHYSICS LETTERS

where

Ps(N) = 2N2/3 +N + 1/3,

which does not depend on t. Also

PN (t) =1

N

N∑i,j=1

Ai csc θi sin(j + 1)θicost+1 θi

1− cos θi.

This last term vanishes with depth (t→∞), while the first is precisely the asymptotic valuefor the pressure. Thus, we have proven that the model predicts arching and calculated Ps;clearly the numerical evaluation of these formulas agrees with the numerical experiments ofthe model.

Continuum analysis. – We now discuss the continuum limit of the averaged equations.If 1 � N , the size of the grains is much smaller than the size of the system and the weightshould not vary appreciably from one site to the adjacent ones, in the horizontal or the vertical

directions; i.e.,∣∣∣w(t)i+1 − w

(t)i

∣∣∣ � w(t)i and

∣∣∣w(t+1)i − w(t)

i

∣∣∣ � w(t)i . The discrete variables for

the horizontal position i and depth t can then be replaced by the continuous variables x andt, such that x ∈ [0, 2N ] and t ∈ [0,∞) [15]. Hence, eq. (1) takes the form

wt =1

2wxx + 1, (6)

where now the subindices denote differentiation (wx ≡ ∂w/∂x). The partial differentialequation, an inhomogeneous heat equation, must satisfy the initial condition w(x, 0) = 1 andthe boundary conditions 2wt(0, t) = 1 + wx(0, t) − w(0, t), and 2wt(2N, t) = 1 − wx(2N, t) −w(2N, t). This parabolic equation is in contrast with alternative approaches that lead tohyperbolic [16] and elliptic [17] formulations. As it stands, this is a generalized Sturm-Liouvilleproblem due to the boundary conditions and we briefly describe the results [18].

The solution can be constructed as the sum of asymptotic and t-dependent parts, w(x, t) =ws(x) + v(x, t). For large t the stationary state is ws(x) = 2(N + 1) + x(2N − x). Thet-dependent part solves a fully homogeneous problem. The solution is

v(x, t) =∞∑n=0

ane−λ2

nt/2

[cosλnx+

1− λ2n

λnsinλnx

], (7)

where λn is the n-th root of the transcendental equation 2λ(λ2 − 1) cos 2λN = (λ4 − 3λ2 +1) sin 2λN . All roots are real and occur in pairs, ±λ0,±λ1, . . .; {λn} is a strictly increasingsequence of positive real numbers. Since 2Nλn/π is not an integer, an appropriate orthog-onalization procedure is required. For N = 50 (200), the first two roots are λ0 = 0.03080(0.007815);λ1 = 0.06160 (0.01563).

The main result is that the weight increases monotonically from 1 to its asymptotic value.The pressure follows from eq. (3), where the sum is replaced by an integral. The asymptoticvalue is Ps = 2N2/3 + 2N + 2. From the first term in eq. (7) the relaxation depth is givenby tr = 2/λ2

0; to leading order tr ' N2, as found before. Janssen’s result corresponds to thisapproximation. Even for rather small values of N , the numerical experiments, the analytical(discrete), and the continuum results fully agree.

Discussion. – The results that stem from the model described above indicate that archingis generic. The simplest model that seems to capture the essential elements to predict and

r. peralta-fabi et al.: arching in confined granular materials 81

characterize arching, is an array of disks in which each one is supported unevenly by those lyingunderneath; those in contact with the surrounding walls lean partially on them. The naturaldisorder in granular systems and the variations in shape and orientation, are incorportad inthe model through the use of uniformly distributed random variables. While more realisticmodels, incorporating adequate contact forces, could give the theory its quantitative value,solid friction at the walls is sufficient to provide the general features of arching.

The model can be extended to account in a simple manner for defects in the column ofgrains by assuming that each site of the lattice has a probability p (p � 1) of being empty.This is done in such a way that no two adjacent sites can be unoccupied. When a site is empty,the two beads on the layer above distribute their weights so that the parts that correspondto the empty site are transferred sideways by the beads until the wall is reached, where theyare absorbed. This mimics the leaning of contiguous beads when a hole is created in theconstruction history of the stack. In this case we find a saturating pressure and a relaxationdepth that are much smaller than those reported here [18]. For close packing, Ps and tr taketheir maximum value.

We numerically modified the boundary conditions so that a fraction α of the weight of thebeads resting on the walls was lost. We found that for fixed N arching always occurs butPs depends on α. On the other hand, for fixed α, Ps/N

2 approaches the value 2/3 as Nincreases [18].

The main conclusion drawn from this model, confirmed numerically and by the experimentsthat we have carried out by random deposition of rough spherical beads, to be reportedelsewhere, is that arching is always present [19]. Also, the asymptotic values, Ps and tr foundwithout holes are the maximum possible values; these then provide a theoretical upper boundthat might be useful from the practical point of view. We have also studied numericallythe three-dimensional version of the model with the same qualitative results, that is, that Ps

and tr scale as N2 [18]. However, Janssen’s analysis indicates that the characteristic depthscales with N , clearly at odds with this model’s prediction. To our knowledge, there is noexperimental evidence on how this length scales nor any theoretical basis besides Janssen’sresult. Otherwise, the functional dependence of the pressure on depth, to first order, coincidesin both Janssen’s analysis and this model. Furthermore, a possible far reaching consequenceof this model, is the fact that its continuum version can be easily extracted as a well-posedmathematical problem. The latter indicates that it implicitly contains the required equilibriumclosure equation.

***

RP-F thanks V. Romero-Rochın and J. Goddard for many profitable and enjoyablediscussions. CM thanks the Fundacion UNAM for an undergraduate scholarship. RR thanksH. Larralde for interesting and enlightening discussions. This work was supported by GrantsUNAM, DGAPA IN-106694, IN-103595, and IN-107197.

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