Where does a cohesive granular heap break?

Transcription

1 Eur. Phys. J. E 14, (2004) DOI /epje/i THE EUROPEAN PHYSICAL JOURNAL E Where does a cohesive granular heap break? F. Restagno 1,a, L. Bocquet 2,b, and E. Charlaix 2,c 1 Laboratoire de physique des solides, UMR CNRS 8502, Bat. 510, Campus universitaire, Orsay Cedex, France 2 Laboratoire physique de la matière condensée et nanostructures, UMR CNRS 5586, Université Lyon I, 43, Bd du 11 Novembre 1918, Villeurbanne Cedex, France Received 19 February 2004 and Received in final form 6 May 2004 / Published online: 29 June 2004 c EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2004 Abstract. In this paper, we consider the effect of cohesion on the stability of a granular heap and compute the maximum angle of stability of the heap as a function of the cohesion. We show that the stability is strongly affected by the dependence of the cohesion on the local pressure. In particular, this dependence is found to determine the localization of the failure plane. While for a constant adhesion force, slip occurs deep inside the heap, surface failure is obtained for a linear variation of the cohesion on the normal stress. Such a transition allows to interpret some recent experimental results on cohesive materials. PACS n Granular systems Gt Powders, porous materials 1 Introduction The economic impact of particle processing is enormous. Better methods for the design and synthesis of unit operations involving divided solids have been identified as a critical need, especially for pharmaceutical, agrochemicals and specialty chemicals. One of the important features in industrial processes is to better understand the transition between a static granular medium and the avalanche process. Avalanches in non-cohesive granular media have been extensively investigated [1, 2]. A common characteristic of these studies is that granular motion occurs in a relative thin boundary layer (around ten grains) at the surface [3] independent of the size of the sample. On the other hand, recent experiments have explored the relatively new subject of cohesive granular media such as humid granular media. The presence of capillary bridges between the grains generate adhesive forces which strongly affect the stability of the samples [4 12]. These capillary bridges either originate in small amounts of added fluid, and this situation corresponds to lightly wet granular media or either are created by a condensable vapour in the atmosphere [7 9]. In the latter case, the vapour is in chemical equilibrium with the interstitial liquid bridges. These experiments concern the so-called moist granular media. As shown by all these experiments, the behaviour of humid granular media strongly departs from the dry materials and many new features appear. In a series of experiments on lightly wet granular media, Tegzes et al. [13], a b c restagno@lps.u-psud.fr lbocquet@lpmcn.univ-lyon1.fr charlaix@lpmcn.univ-lyon1.fr measured the angle of repose of a granular medium (by the draining crater method) as a function of the liquid content and of the size of the system. They observed an increase of the angle of repose with the liquid content. More precisely, they obtained three regimes: i) at low liquid content, the granular regime, the avalanche occurs at the surface and the angle of repose does not depend on the size of the system; ii) at intermediate liquid content, the correlated regime, the avalanche takes place more deeply in the heap, the angle of repose depends on the size of the system; iii) at high liquid content, a kind of plastic flow is observed. These various regimes are not totally accounted for by previous theoretical analysis. In numerical and analytical modelisations, cohesion is usually taken into account as a constant adhesion force. To cite a few examples, Forsyth et al. [14] and Valverde et al. [10] used a constant Van der Waals force to interpret their experimental results, Olivi-Tran et al. [15] used a constant capillary force or a constant force due to a solid bridge in particle dynamics (PD) simulations, Albert et al. [4] used a constant capillary force in a geometrical model and Nase et al. [16], Mikami et al. [17] used a constant capillary force in PD simulations. F. Radjai [18] studied the cohesive granular texture by numerical simulations. Schulz et al. [19] presented a numerical study of a shear-induced solid-fluid transition in wet granular matter. The simulation is based on a simple model where dissipation is assumed to be entirely due to the hysteretic character of a constant cohesive force. In this paper we show that the assumption of a constant adhesion force is not justified in many practical cases. We then show that the dependence of the adhesion

2 178 The European Physical Journal E on normal stress, usually omitted in the literature, is a key point in determining the stability of a cohesive granular heap. This effect leads to a dependence of the stability on the system size, as found experimentally. To this end, we shall use a continuum analysis [20] to study the stability of cohesive sandpiles taking into account a non-constant cohesive stress in the pile. The paper is divided in two parts. First, we briefly show that in many situations, the adhesion force between the grains depends on the normal stress. Then, in the second part, we show how this relationship affects the stability of a heap and the localization of the slip plane upon failure. 2 Contact forces: A brief review When two grains comes into contact, an interaction force builds up. The latter can be separated into various components with different physical origins: a direct (elastic) repulsion of the solids, normal to the surfaces in contact; a tangential solid friction force; and a normal adhesion force. The experimental and theoretical characterization of these surface forces has been the object of extensive studies for more than 50 years in the field of tribology, pioneered by several important scientific schools around D. Tabor [21], B. Derjaguin [22] or K. Johnson [23]. Let us first recall the results for the adhesion force between two perfectly smooth spheres in contact. This problem has been addressed theoretically some years ago [22 24]. The adhesion force F S-S in this simple case is then F S-S = fπγr, (1) where R is the sphere radius, γ is the solid/interstitial medium surface tension (γ = γ SG if the spheres are in a gas atmosphere, and γ = γ SL if they are immersed in a liquid), and f is a numerical factor between 1.5 and 2. This equation is valid if the roughness of the two surfaces is much smaller than the range of the interaction forces between the surfaces. In the case of short-range adhesion forces, this means atomically smooth. An obvious feature of this result is that adhesion does not depend on the pressure applied on the surfaces before separating them [25]. Indeed measured values of the adhesion force between real surfaces generally departs from equation (1) because real surfaces are not smooth [21]: the area of real contact between real spheres is much smaller than the apparent contact area due to the surface roughness, leading to qualitative modifications of the adhesion force. We will now show how we can understand the existence of load-dependent adhesion forces between surfaces due to surface roughness in real granular materials, such as reported results in the literature [21, 26 28]. One may separate two different classes: the case of adhesion due to (direct) short-range interactions; and the case of adhesion due to external binder, such as capillary bridge between the grains. R 2a 0 F (a) δ Fig. 1. (a): contact between two perfect spheres of radius R, Young modulus E. (b):contactbetweentworoughparticlesof radius R. 2.1 Adhesion due to short-range forces The case of rough surfaces with direct short-range interactions has been considered experimentally by Restagno et al. [28]. The adhesion force between a sphere and a plane, both made of weakly rough pyrex, has been measured using a surface forces apparatus. They obtained a load-dependent adhesion between the surfaces, F adh, which scaled with the normal load F,asF adh F 1/3. This result provides a good example to exhibit the effect of the underlying roughness. Indeed, in contrast to the perfectly smooth surfaces, mechanical contact occurs only at the tip of the surface asperities, and the real contact area A r is accordingly much smaller than the apparent contact area A app (Fig. 1). Equation (1) must be therefore modified by replacing the real surface tension by an effective surface tension γ eff = γa r /A app. Now, the area of real contact is assumed to be proportional to the normal load of the surfaces [21]: (b) A r F. (2) This widely recognized result is usually explained in terms of plastic deformation of the spherical asperities [21], or on the basis of elastic deformation of a statistical assembly of asperities [29]. On the other hand, the apparent contact (large scale) area is still given by the elastic contact area as calculated by Hertz [30] : A app (FR/E ) 2/3, with E = E/[2(1 ν 2 )] the reduced Young modulus of the material, defined in terms of the Young modulus E and the Poisson modulus ν of the material. This leads eventually to F adh F 1/3, in agreement with the experimental results. This first example shows that, in contrast to perfectly smooth surfaces, roughness leads to load-dependent adhesion forces. 2.2 Adhesion due to capillary bridges In the previous experiments, the cohesion stems from the direct interaction between the two (rough) surfaces. However, adhesion might also originate in the presence of liquid bridges between the grains. This problem has been

3 F. Restagno et al.: Where does a cohesive granular heap break? 179 studied between two grains by Willet [31]. In many practical situations, the presence of liquid bridges connecting the grains leads to strong adhesive forces [4 7,9,8]. Two different situations are generally reported: i) the granular material is placed in a humid atmosphere; ii) a small amount of liquid is added to the grains. When the material is placed in an atmosphere of a condensable vapour [7,8], the size of the liquid bridge is fixed via chemical equilibrium by the value of external humidity. The size of such bridges is usually very small and fixed by the Kelvin radius r K, i.e. around a few nanometers. The liquid bridges therefore condense only in the regions close to real contact and the wetted area, A w, is accordingly expected to be proportional to the real contact area, A r. Now, the adhesion force between the grains is proportional to A w, and one gets eventually F adh γ r K A w. Since, as discussed above in equation (2), A r is proportional to the external load, one gets eventually a linear dependence of the adhesion force on the external load: F adh f(r K )F. This result relies however on few assumptions, which we quote here. First, it is restricted to small liquid bridges, such that only the asperities in contact are wetted by the liquid bridges. This amounts to assume that the size of the liquid bridge, typically of the order of the Kelvin radius, is smaller than the asperity, or roughness, size. In the other limit, i.e. when the liquid completely fill the asperities, roughness is not relevant anymore and the adhesion force reduces to the previous expression in equation (1), F adh = fπγr. This is obtained when the condensed volume V cond, which is a complicated function of the vapour pressure, reaches the limiting value: V cond >l 2 RR, where l R is the typical size of the asperities. Another underlying assumption is that the material hardness is larger than γ the Laplace pressure, r K, of the order of Pa. In the other limit, which might occur, e.g., for polymers or ductile metals like lead, an interplay between the adhesion force and the plastification of the asperities might occur, yielding a complex adhesion force-normal load dependence. We do not explore this limit, which is beyond the scope of the paper. The second situation corresponds to the problem of a granular material with a small amount of added liquid. The physical parameter is now the volume of added liquid. A specific difference to the previous case of a condensable vapour is that the size of the liquid bridge is general larger when liquid is added to the material. This situation has been studied extensively by Halsey and Levine [32], however, in the limiting case of rigid surfaces. They obtained therefore an adhesion force independent of the load between the grains, but dependent on the liquid content. Two regimes were discussed. The first regime is the socalled rough regime in which the liquid gradually fills the rugosity. This picture is however valid when the size r of the bridges is lower than a typical scale of roughness, l R. In the opposite case, r>l R, roughness is not pertinent anymore and the capillary force reduces to a constant value 2πγR, with R the radius of the beads, and γ the liquid vapour surface tension. The case of a deformable surface has not been considered to our knowledge. As above, an interplay between the adhesion force and the plastification of the asperities might occur, a complex situation in itself which we do not discuss specifically in this paper. 2.3 Discussion To sum up, our main point in this short review is to point out that a normal load dependence of the adhesion force is expected in the case of real surface. In general, one may write the adhesion force between two surfaces as F adh F n (3) with various expected exponents n. For humid granular materials, a transition from n =1ton = 0 is expected, depending on the relative size of the capillary bridges compared to the roughness scale: For a small amount of liquid (or low humidity), the value n = 1 is expected, while a large amount of liquid (or close to 100% of humidity) corresponds to n = 0. On the other hand, for dry or liquid saturated materials in which cohesion is expected to originate in direct (e.g., Van der Waals) interactions, a fractional value n = 1/3 is predicted. In order to discuss the stability criterion of a cohesive heap, we now derive the local cohesion stress from the adhesion force. In the following, we will call adhesion force a force which sticks two grains together and cohesion the resulting stress in the material due to the adhesion force. 3 Failure of a cohesive heap 3.1 From adhesion force to local cohesion stress The problem of the derivation of the interparticle forces F from the bulk stress σ is an old one. For a system of hard monodisperse spherical particles with a random isotropic packing, a continuum-like picture leads to a simple linear relationship between force F and stress σ: σ = φk F, (4) 4πR2 where R is the radius of the particles, k is the coordination number, which is defined as the average number of contacts per particle, and φ is the volume fraction. Such a relationship has been widely used in the literature for powders or wet granular media [11,33]. From equation (4), we can deduce the relationship between the adhesion force F adh and the adhesive stress c adh in the granular medium: c adh F adh R ; and the relation between the normal load and 2 the normal stress c F R. Assuming a general 2 relationship of the form of equation (3), one gets therefore an adhesive stress which depends on the normal stress, as shown in Figure 2, c adh = c 0 ( σ σ 0 ) n, (5) where n, σ 0 and c 0 characterize the properties of the material. This relationship is our general starting point to study the stability of a cohesive heap.

4 180 The European Physical Journal E c n > 1 "Apparent non-cohesive regime" n = 1 "Granular regime" H c 0 0 < n < 1 "Size dependent avalanche regime" n = 0 "Classic adhesion regime" θ α L σ 0 σ Fig. 3. Geometry of the granular heap. θ is the actual slope of the heap, while α locates the position of a possible failure. Fig. 2. Different regimes of cohesion in a granular heap. 3.2 From cohesion to failure The most simple approach in the continuum limit to study the failure of a granular material is the plastic theory of Mohr-Coulomb. The validity of the continuum approach has been widely discussed. Nevertheless, continuum approaches allow to predict the Green function of a granular layer [34] and are widely used in soil mechanics [20, 33]. We consider the stability of a cohesive granular heap, characterized by an adhesive stress depending on the normal stress, as given in equation (5), in which we suppose the Mohr-Coulombcriterion valid. Our aim is to locate in such a material the slip plane, where failure occurs. The basis of our analysis is the Coulomb criterion: a granular material is stable if for each surface inside the material the following inequality is obeyed: τ µσ, (6) where τ is the shear stress and σ is the normal stress. For a cohesive material, we assume that this condition can be generalized by adding in equation (6) the adhesive stress c adh, as defined in the previous section, to the normal stress σ [7]. We consider the geometry depicted in Figure 3: A granular heap, with height H, makes an angle θ with the horizontal. The heap is supposed to be invariant in the direction perpendicular to the figure. We follow and generalize the approach given in reference [20] to the case of a cohesive material with the law of cohesion given in equation (5). In the simplified description of reference [20], the slip surface is assumed to be planar, making an angle α against the horizontal. The force balance in the direction perpendicular and parallel to the slip plane leads to the following conditions: F = P sin α, N = P cos α + F adh, (7) where P is the weight of the granular material above the slip plane; c = F/S slip and σ + c adh = N/S slip are the normal and tangential stress along the possible slip plane with area S slip. In these variables, the stability criterion, equation (6), writes F<µN. A simple geometric calculation leads to the relationships [20]: P = 1 ( 1 2 ρglh2 tan α 1 ), tan θ S slip = HL (8) sin α with L the length of the heap in the invariant direction. Gathering these results, one gets the following stability criterion: 1 2 ρgh2 ( 1 tan α 1 ) (sin α µ cos α) µ c adh H tan θ sin α. (9) Introducing the angle θ 0 defined as tan θ 0 = µ, this relation can be adequately rewritten as sin(θ α)sin(α θ 0 ) 2sinθ 0 sin θ c adh ρgh. (10) Now, one has to introduce the dependence of the cohesive stress c adh as a function of the normal stress, as discussed in the previous section. We shall use the general expression given by equation (5). Using σ = P cos α/s slip, one gets after some algebra ( ) n ( ) n ρgh sin(θ α)cosα c adh = c 0. (11) 2σ 0 sin θ Introducing the cohesion parameter ζ coh defined as ζ coh = c 0 ρgh and the function f[α] defined as f[α] = equation (10) rewrites ( ) n ρgh, (12) 2σ 0 (sin(θ α))1 n (cos α) n sin(α θ 0 ), (13) f[α] ζ coh sin θ 0 (sin θ) n 1. (14) The heap is therefore stable at an angle θ if this inequality is verified for all possible α (with 0 α θ). In the opposite case, when this inequality is not verified for some values of α, the slip plane corresponds to the value of α for which this inequality is first violated. Note that for a non-cohesive material, ζ coh = 0, the maximum angle of stability is simply θ n 1 case First, if θ θ 0, the function f[α] is always negative and the stability condition is trivially verified: For θ<θ 0 the heap is always stable.

5 F. Restagno et al.: Where does a cohesive granular heap break? 181 Fig. 4. Plot of f[α] as a function of α, for aspecific choice of parameters: θ =60, θ 0 =30 and n =1/3. The dashed line locates the position of the maximum α m. The opposite case θ > θ 0 is more complex. A typical plot for this function is then given in Figure 4. This function does exhibit a maximum for a value α m,verifying f/ α(α m ) = 0, and depending on the heap angle θ. According to criterion (14), the heap is stable if f[α m ] ζ coh sin θ 0 (sin θ) n 1. The maximum angle of stability θ m thus is solution of the implicit equation f[α m ]=ζ coh sin θ 0 (sin θ m ) n 1. (15) A straightforward calculation shows that α m obeys the following relationship: sin(2α m θ 0 θ)cosα m = n cos(2α m θ)sin(α m θ 0 ). (16) It is interesting to consider the two limiting cases n =1 and n =0. For n = 1, the solution of equation (16) is α m = θ; and θ m is found to obey sin(θ m θ 0 ) = ζ coh. (17) cos θ m sin θ 0 This relationship can be rewritten in a more explicit form as ( tan(θ m )=µ(1 + ζ coh )=µ 1+ c ) 0 (18) 2σ 0 with µ = tan θ 0. This case presents two important characteristics: i) failure occurs here at the surface of the heap, as emphasized by the relationship α m = θ; ii) whatever the heap height H, thereisalwaysaneffect of cohesion on the maximum angle of stability. This regime has been called the granular regime according to Tegzes experiments in which a cohesion effect on the avalanche angle is found without dependence on the heap size. In Tegzes experiments, this regime is found at low liquid content. In Section 2, we Fig. 5. Dependence of the maximum angle of stability, θ m (solid line) and failure plane location, α m (dotted line), as a function of the cohesion ζ coh (angles are here given in degrees). Physical parameters are: θ 0 =30, n =1/3. The dashed line is the approximate solution, as given by equation (20). showed that the value n = 1 is indeed expected when cohesion results from small capillary bridges between surfaces asperities. For n = 0(classic adhesion regime, i.e. constant adhesion force), the solution of equation (16) is α m = (θ 0 + θ)/2, and θ m verifies the equation 1 cos(θ m θ 0 ) 2sinθ sin θ 0 = ζ coh = c 0 ρgh. (19) This is a classical result, as obtained, e.g., in reference [20]. It is important to note that in this case, i) the heap fails deep inside the material, since α m <θ m ; ii) the maximum angle of stability θ m depends on the height of the heap H. Such a dependence is in fact expected when one realizes that for n = 0, a capillary length scale can be defined on dimensional grounds: l cap = c 0 /ρg (see, e.g., Eq. (12)). This capillary length gives the size of a macroscopic element of the granular medium whose weight equals the adhesion force which act on it. For instance, it maybe the maximum size of a powder aggregate which can remain stuck under a horizontal surface. Equation (19) shows that the maximum angle of stability of the heap reduces to θ 0 when the size of the heap is very large compared to the capillary length. For n in between these two limiting cases, one has to solve numerically equations (16) and (15). α m lies in between θ and (θ+θ 0 )/2. We show in Figure 5 a typical result for the dependence of θ m and α m as a function of ζ coh for n =1/3. The dependence of θ m in the limit of small cohesion can be computed analytically. For ζ coh = 0, one has α m = θ m = θ 0 and one may expand the angles around this values for small ζ coh : α m = θ 0 + δα and θ m = θ 0 + δθ. First, linearizing equation (16),

6 182 The European Physical Journal E one gets the relationship δα = δθ/(2 n). Introducing this condition into equation (15), one gets eventually θ m θ 0 = γζ 1 2 n coh (20) with γ =(2 n)/(1 n) (1 n)/(2 n) sin θ 0 (cos θ 0 ) n/2 n. Inserting the H-dependence of ζ coh, as defined in equation (12), one gets, therefore, θ m θ 0 H n 1 2 n. (21) This shows, in particular, that the dependence of the maximum angle on the heap height is a direct measure of the power n, characterizing the normal stress dependence of the cohesion stress. This regime is called the size-dependent avalanche regime n>1 case In this case, it is easy to show that whatever the cohesion, the maximum angle of stability is θ 0. This regime is called the apparent non-cohesive regime. First, as in the previous case, when θ<θ 0, the condition of stability, (Eq. (14)) is trivially obeyed since the function f[α] is negative. Now, for θ>θ 0, f[α] goes to infinity when α θ. The heap is therefore always unstable, whatever the cohesion. As a result, θ = θ 0 is the maximum angle of stability, independently of the cohesion ζ coh. 4 Discussion In this paper we have studied the effect of an adhesive stress on the localization of the slip plane in an avalanche process. More precisely, we have shown that in a lot of practical cases, the adhesive stress in a material depends on the normal stress. We have calculated, using a Mohr- Coulombanalysis, the internal angle of slip α m. A few conclusions can be drawn from these results: First, a change of regime in the cohesion, e.g. a change from n = 1 (linear dependence of the cohesion stress with the normal stress) to n = 0 (constant cohesion) as discussed in Section 2, will change dramatically the localization of the slip plane: For example, while for n = 1 slip occurs at the surface of the the heap (α m = θ m ), it will fracture in the interior of the heap for n =0(α m <θ m ). In other words, any change of slip behaviour reflects a transition of cohesion regime. Second, the maximum static angle, θ m, does depend on cohesion via the dimensionless ( ) parameter, ζ coh, defined as ζ coh = c0 ρgh n. ρgh 2σ 0 An important remark is that for n 1, the maximum static angle depends on the height of the heap, H. On the other hand, for the specific value n = 1 which is expected in some physical situations (see discussion in Sect. 2) this dependence disappears and θ m is an intrinsic property of the material, independently of the geometry. This features provide a framework to discuss the results of Tegzes et al. [13]. We emphasize that since Tegze s et al. used a cylindrical geometry and studied the angle of repose measured with the draining method, we cannot expect to obtain a detailed quantitative analysis. Nevertheless, in view of our results, it seems natural to interpret the different regimes observed in the experiment, as an indication of a change in the cohesion regime. In these experiments, they measure the repose angle as a function of the added liquid in the sample (oil). This added liquid is measured in terms of an averaged oil film thickness on the granular beads. Their observations can be summarized as follows: At low liquid content (corresponding in Tegzes experiments to δ liq < 20 nm), the avalanche takes place at the surface of the heap and the angle increases linearly with the liquid content. Their data for small angles are compatible with a linear dependence, tan θ r = tan θ 0 + κδ liq with θ 0 =20 and κ = nm 1. While such a linear dependence could be interpreted in terms of the Hasley-Levine model (in their roughness regime ), the latter analysis also predicts that the heap should fail at depth in the presence of cohesive forces [32], rather than at surface, as observed by Tegzes et al. On the other hand, a surface failure is indeed expected within our approach in the granular regime, where the cohesion stress is proportional to the normal stress (corresponding to a n = 1 exponent). As noted above, this regime is indeed expected at low liquid content when the capillary bridges do not fill the full interstitial space between the grains. This therefore suggests that the fact that the Halsey-Levine description does not capture the surface failure localization is due to their assumption of infinitely rigid spheres, which leads to a cohesion independent of normal stress. The experimental result of Tegzes also suggests that the prefactor c 0 in the cohesion stress, see equation (5), depends linearly on the added liquid content in this regime. More specifically, if we compare their results with equation (18), we do obtain µ =0.36 which is a classical value for spherical beads, and F adh /F = c 0 /σ 0 =2κδ liq /µ = λδ liq, with λ and δ liq is expressed in nanometers. Intuitively, one expects to recover such a dependence by generalizing the Halsey- Levine model, taking into account the elasto-plastic properties of the grains. We leave this (difficult) point for future studies. At larger liquid content (δ liq > 30 nm), failure occurs via bulk avalanches. Moreover, a dependence of the avalanche angle with the size of the container is observed. Again, these observations can be accounted for within our model in a regime with a cohesion depending on normal stress (i.e. with an exponent n<1). At very large liquid content, the repose angle reaches a saturation value, corresponding to an adhesion force between two grains independent of the liquid content. As discussed above, such a dependence is indeed expected for large amounts of added liquid, when the

7 F. Restagno et al.: Where does a cohesive granular heap break? 183 liquid film overcomes the roughness of the beads. The cohesion stress becomes independent of the normal stress and n = 0. Note that in this limit the predictions of the Halsey-Levine model are identical to ours. This discussion clearly suggests two directions of work in the future. First, from the theoretical point of view, a generalization of the Halsey-Levine model is needed, which takes into account the elasto-plastic deformations of asperities of (rough) grains at contact. This is a difficult task, but it is a necessary step in order to obtain a full picture of the effect of adhesion on granular stability. Second, from the experimental point of view, a more thorough study of the failure plane localization would be required to get information on the specific properties of adhesion forces between grains. In particular, the heap height dependence of the maximum angle of stability should provide a direct measure of the cohesion properties. We therefore hope that this work will motivate further experimental investigation on the stability of cohesive granular materials. In particular, a comparison between different geometries should be really helpful in giving a full picture of the underlying cohesion mechanisms. A particularly interesting geometry is the cylindrical bunker ( Janssen s problem ), in which cohesion effects, as discussed here, are expected to play an important role. We thank the Région Rhône-Alpes for financial support (Programme Emergence ). References 1. H.M. Jaeger, C.-H. Liu, S. Nagel, Phys. Rev. Lett. 62, 40 (1988). 2. J. Rajchenbach, Phys. Rev. Lett. 65, 2221 (1990). 3. H. Jaeger, S.R. Nagel, Science 255, 1523 (1992). 4. R. Albert, I. Albert, D. Hornbaker, P. Schiffer, A.L. Barabási, Phys. Rev. E 56, R6271 (1997). 5. R. Albert, M. Pfeifer, A.L. Ba ra bási, P. Schiffer, Phys. Rev. Lett. 82, 205 (1999). 6. D. Hornbaker, R. Albert, I. Albert, A.-L. Barabási, P. Shiffer, Nature 387, 765 (1997). 7. L. Bocquet, E. Charlaix, S. Ciliberto, J. Crassous, Nature 396, 735 (1998). 8. C. Ursini, F. Restagno, G. Gayvallet, E. Charlaix, Phys. Rev. E 66, (2002). 9. N. Fraysse, H. Thomé, L. Petit, Eur. Phys. J. B 11, 615 (1999). 10. M. Valverde, A. Castellanos, A. Ramos, Phys. Rev. E 62, 6851 (2000). 11. M. Quintanilla, J. Valverde, A. Castellanos, R. Viturro, Phys. Rev. Lett. 87, (2001). 12. D. Geromichalos, M.M. Kohonen, F. Mugele, S. Herminghaus, Phys. Rev. Lett. 90, (2003). 13. P. Tegzes, R. Albert, M. Paskvan, A.L. Barabá si, T. Vicsek, P. Schiffer, Phys. Rev. E 60, 5823 (1999). 14. A.J. Forsyth, S.R. Hutton, M.J. Rhodes, C.F. Osborne, Phys. Rev. E 63, (2001). 15. N. Olivi-Tran, N. Fraysse, P. Girard, M. Ramonda, D. Chatain, Eur. Phys. J. E25, 217 (2002). 16. S. Nase, W. Vargas, A. Abatan, J. McCarthy, Powder Technol. 116, 214 (2001). 17. T. Mikami, H. Kamiya, M. Horio, Chem. Eng. Sci. 53, 1927 (1998). 18. F. Radjai, I. Preechawuttipong, R. Peyroux, in Continuous Modelling of Cohesive-Frictional Materials, editedbyp.a. Vermeer, S. Diebels, W. Ehlers, H.J. Herrmann (Springer, Berlin, 2001) pp M. Schulz, B.M. Schulz, S. Herminghaus, Phys. Rev. E 67 (2003). 20. R. Nedderman, Statics and Kinematics of Granular Materials (Cambridge University Press, Cambridge, 1992). 21. F. Bowden, D. Tabor, The Friction and Lubrification of Solids (Clarendon Press, Oxford, 1950). 22. B. Derjaguin, V. Muller, Y. Toporov, Colloids Surf. 7, 251 (1983). 23. K.L. Johnson, K. Kendall, A. Roberts, Proc. R. Soc. London, Ser. A 324, 301 (1971). 24. D. Maugis, J. Colloid Interface Sci. 150, 243 (1992). 25. J. Israelachvili, Intermolecular & Surface Forces, 2ndedition (Academic Press, London, 1992). 26. R.H. Schmidt, G. Haugstad, W.L. Gladfelter, Langmuir 19, (2003). 27. S. Biggs, G. Spinks, J. Adhes. Sci. Technol. 12, 461 (1998). 28. F. Restagno, J. Crassous, C. Cottin-Bizonne, E. Charlaix, Phys. Rev. E 65, (2002). 29. J. Greenwood, J. Lubric. Tech. Trans. ASME 1, 81 (1967). 30. L. Landau, E. Lifchitz, Élasticité (Mir, Moscou, 1982). 31. C.D. Willett, M.J. Adams, S.A. Johnson, J.P.K. Seville, Langmuir 16, 9396 (2000). 32. T.C. Halsey, A.J. Levine, Phys. Rev. Lett. 80, 3141 (1998). 33. T. Mason, A. Levine, D. Ertas, T. Halsey, Phys. Rev. E 60, R5044 (1999). 34. G. Reydellet, E. Clement, Phys. Rev. Lett. 86, 3308 (2001).