rheology of granular media

ELSEVIE R Current Opinion in Colloid & Interface Science 4 (1999) 294-299 www.elsevier.nl/locate/cocis

Rheology of granular media

Eric Clement * Laboratoire des Milieux Disordonn6s et Hitirogknes, Universiti Pierre et Marie Curie-UMR 7603 du CNRS, Boite 86-4, Place Jussieu,

75252, Paris, France

Abstract

This review is concerned with the physics of dry granular media which is the focus of many recent studies. One reason for this interest is not only due to its practical importance but also, because it is currently seen as a fundamental model with a rather broad range of applicability. The concepts and questions developed in this field have general implications in other fields such as the rheology of complex fluids (dense pastes or colloidal suspensions), the mechanics of various composite solids such as reinforced rubber, many geophysical situations and even the nano-tribology of molecular films ... We review some of the aspects that were most recently discussed and which are under the scope of an intense debate. 0 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Granular materials; Force chains; Arching; Avalances; Silo; Hopper; Solid friction; Sand pile

1. Introduction

Understanding the physics of granular media is an old problem. At the end of the 18th century Coulomb had proposed that the maximum angle supported by a sand heap could be understood in analogy with the friction properties of two solid bodies in contact. The angle of repose of a sand pile is a current manifesta- tion of a fundamental material quantity a, called the internal angle of friction [l"] which is analogous to the coefficient of solid friction [2], p, = tan@ (see Fig. la,b). There are many other important manifestations of this striking analogy with solid friction. For example, when a granular assembly is compressed, both horizontally and vertically, a very well defined and localized shear band occurs when the ratio of the tangential force to the normal force with respect to a sliding plane reaches a value almost equal to the tangent of the internal angle of friction [52] (see Fig.

* Tel.: + 33-1445-274434. E-mail address: [email protected] (E. Clement)

lc). Above this threshold one may consider, in a first approximation, that a plastic limit is reached. For a real granular assembly, the previous idea of a unique coefficient determining the internal angle of a granular assembly and a purely plastic behavior must be somehow revised. A dependence of the stress tensor and the internal angle of friction with deformation and compacity seems to hold empirically (see Fig. 2) and this is the base for the standard soil mechanic theories [loo]. This approach is, so far, the most elaborated and sophisticated vision of granular rheology, which applies in the static and quasi-static limits. Unfortunately, it is based on effective relations between stresses, deformations and directions of deformation obtained in an empirical way. For practical purposes, the constitutive elements of the theory are obtained by standard compression tests called triaxial tests, which are often difficult to conduct and the constitutive relations hence obtained, are mostly non-linear, piece-wise, anisotropic, their outcome de- pends crucially on the history of the loading/unload- ing cycles. This point of view is suited for numerical calculations of practical engineering situations and

1359-0294/99/$ - see front matter 0 1999 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 0 2 9 4 ( 9 9 ) O 0 0 0 4 - 7

E. Clement /Current Opinion in Colloid & Interface Science 4 (1999) 294-299 295

W+ 6 ..,..... .. ..

Fig. 1. Analogy of the Mohr-Coulomb yield criterion with the laws of solid friction: (a); solid block on a plane, at the onset of motion. The reaction force opposes the weight W such that: T / N = p =

tan@. (b); the limiting slope for a sand pile @ is such that any extra granular material (hashed surface) will slide down in analogy with (a). The angle @ is the internal angle of friction of the granular assembly. (c); schematic representation of a triaxial compression test when a shearing band occurs (with an angle a), the normal (a) and tangential (7 ) stresses satisfy the relation T/U = tan@.

does not give any explanation, on a physical basis, for the observed phenomena.

Another limit for the rheology of granular assemblies was investigated in the 1980s and concerns rapid granular flows. The standard vision borrows the theory of dense gases [3] and adapt it to a situation where the momentum transfer and energy dissipation occurs through sequences of binary dissipative colli-

U

E, 0

L v

x .. *. **.... .%.

'..*W *. . . . . . . . . . . . . , . . . . , , .

Fig. 2. Schematic representation of the internal failure properties for real granular assemblies. The direction of strain variations is positive. (a): shear-stress T as a function of strain y for compacted sand (solid line) and loose sand (dotted line). (b): volumetric strain E , (compacity variations) as a function of strain y for compacted sand (solid line) and loose sand (dotted line). The expansion effect (ev > 0) associated with a stress barrier for dense assemblies is called the Reynolds dilatancy effect.

sions. Hydrodynamic equations are derived under a closed form [3,41. This approach is somehow success- ful when the system stays in the collisional limit. Unfortunately for many real situations, dense structures appear [5], and eventually the granular materials will flow in a state with enduring contacts and this approach is bound to fail.

2. Arch formation and stress paths

Granular material is made of grains which interact mainly via contact forces. An important property is the lack of resistance of contact forces to any exten- sion (for non-adhesive grains). Generally, one uses both local compressive elasticity (which can be taken to the hard-grain limit) and solid friction to describe the contact between two grains. This review is concerned mainly with dry assemblies, typically with grains of approximately 1 mm or more. Experiments on model granular piling show that the distribution of contact forces in such a system is strongly disordered since one observes a distribution of large forces paths encompassing regions bearing relatively small loads [6'] (see Fig. 3). It is still a problem to understand how this very heterogeneous distribution of forces would affect macroscopic quantities such as stresses and deformations. Recent experiments [7",81 and computer simulations [9-121 have studied how the distribution of contact forces in a granular assembly fluctuates spatially. The typical standard deviation is on the order of the mean force and the large forces probabilities decay exponentially. In real experimental systems, the small force probabilities [7",81 are almost equal to the probability of the average force. This can be seen as a characteristic feature of the fragile aspect of granular material, which means that a subsequent amount of contacts bears little load and can easily be removed, hence changing the topology of the force paths. The self-organization properties of the force network was investigated rather finely [13°00]. Generally, the mesh-size of the large force network is approximately 10 grain.

3. Static granular assemblies

Granular assemblies being at first glance both liquid-like (they flow) or solid-like (a heap exists), their static mechanical behavior can be seen as a part of a general interrogation on the rheology of granular flows. Below the onset of motion, an analogy is currently made with the Coulomb model of friction. The analogous of this static criterion is called the Mohr-Coulomb criterion [l"] and it holds at rest for

296 E. Clement /Current Opinion in Colloid & Interface Science 4 (1999) 294-299

4 Fig. 3. Schematic representation of the force distribution network in a compact granular assembly during a two-dimensional compression experiment. The larger the contact forces, the thicker the lines between the grain centers.

normal and tangential stresses (respectively, u and T) with respect to any orientation of an eventual slip plane.

171 < t a n a u (1)

In this frame work there is no fundamental theory able to define the mechanical status of a granular assembly at rest. Classical soil mechanic approaches could, in principle, propose an answer, but require information on the whole deformation history of the granular assembly from a reference state which is quite hard to define [14"1. The fact that mesoscopic structures exist and may bias the propagation of contact forces is at the origin several conjectures relating the direction of propagation for stresses to the local shearing stress. This can be seen as the macroscopic emergence of a vault effect capturing the underlying anisotropy of the contact force network [15-17*0*1. General considerations on symmetry and the fact that no elastic deformation scale should be considered to describe a hard grain limit, could imply a relation between tress tensor components of this type

which is assumed to hold below the Mohr-Coulomb limit. Relation (2) relating the stress tensor components uij, is given here in two-dimensions for simplic- ity. One example of such a relation is called the Oriented Stress Linearity model (OSL) [17***1. This type of relation between stresses has started a heated debate, the reason being that the nature of the stress equations is then of the propagative type (i.e. hyper- bolic). More standard mechanical approaches provid- ing relations between stresses and deformations have the character of elasticity equations (i.e. elliptic) below the plasticity threshold (these models are called elastoplastic). Another crucial difference lies in the influence of the boundary conditions. In a propagative equation, just a part of the boundary conditions have to be specified (e.g. stresses at a free surface) and the other boundary conditions must adapt (e.g. the con- tainer walls). In the other interpretation scheme, all the boundary conditions (stresses and strains) must be satisfied at the same time and they define without ambiguity the field values in the bulk [18**]. A tentative to bridge these notions and also to make analogy with the mechanical behavior of colloids in a very concentrated state was proposed by Cates et al. [19"]. These models explain very easily [20] some experimental facts like a pressure dip below a sand pile [51] or the overshoot of pressure at the bottom of a granular column with an overload [211. On the other hand, they do not provide a rigorous derivation for the constitutive coefficients used (that could be seen so far, as mere fitting parameters) and also their necessary relation with compacity. Interesting experimental evidence for the crucial influence of com- paction on the average value of the stresses was performed by Honvarth et al. [22]. Moreover, effects

E. Clement /Current Opinion in Colloid & Interface Science 4 (1999) 294-299 291

like coherent sound wave propagation in sand are hard to conciliate [23] with this approach.

4. Quasi-static flows

When the limit given by the Mohr-Coulomb criterion is reached for a given direction, the granular material starts to flow. Deformations are generally limited to several grain sizes and they form what is called a shear-band [24]. The average density inside a shear band is rather reproducible [251 and soil mechanic engineers call this a 'critical state' [l]. A model experiment probing the apparition of shear bands was carried recently by Ngadi et al. [26"] in a two-dimensional model system and by Miller et al. [27"] in a three-dimensional Couette cell. Large fluctuations in the resistance force were observed with peak values of up to one or two orders of magnitude larger than the average driving force. These large fluctuations indicate the presence of complex mechanical structures such as vaults. In a two-dimensional Couette cell, the shear bands and the associated structure of the force chains were visualized directly using bire- fringence properties of plastic discs [28]. For the same display, a claim was made on the presence of a second order-like phase transition when the packing fraction of discs is lowered: the average chain length diverges and the force distribution changes drastically [29].

At very low shearing rates, the yield dynamics of the granular assembly show that the striking analogy with solid on solid friction can be carried further. A 'stick-slip' instability is also observed for shearing devices which low stiffness [30",31"]. It means that a dynamic friction coefficient can empirically be defined [30"1. According to the conditions of solicita- tion, the nature of the grains, one may observe a creep response prior to large slip events [30"] or an amplification of the resistance force leading to a large distribution elastic energy release [31"]. At a higher shearing rate or with a stiffer shearing apparatus, the stick-slip dynamics disappears and a behavior closer to a plastic response is recovered. The friction force exerted on a rod by a moving granular flow was also studied in the same experimental range [32].

5. Onset of avalanches - surface gravity flows

The passage from a solid-like behavior to a liquid- like flow can be achieved by increasing the slope of a sand pile. If the rate of increase is small, a discrete series of avalanches is triggered with a peaked mass distribution [33,34]. In a rotating drum more complex dynamics are observed such as intermittent dynamics

at a low rotation rate [35] and at a higher rotation rate a continuous flow regime [361. A recent pheno- menological model for the onset of flow was proposed based on a dynamical interaction between a flowing phase and a rest phase [37]. In the early stage of this theoretical scheme, the flowing layer is supposed to be uniform which could apply only to ultra-thin flows. Experimentally, there is evidence that a flowing granular layer is thicker though limited in size (at a laboratory scale approximately 5-15 grains deep) [38,39"']. This ingredient was also taken into account [40]. In reality a granular flow would also ex- hibit a vertical velocity gradient [38,39"'], which should be taken into account [41,42]. Experiments on granular flow down a rough inclined plane were performed, and show that the surface roughness modifies crucially the angle for the flow onset. A well-defined relation exists between this angle and the layer thickness called the stopping height, Hstop [431. The thin- ner the layer, the larger the flow onset angle. On such a rough plane, for a range of angles, a steady uniform flow occurs [44] and the surface velocity, U, is measured for surface roughness, different beads and flow thickness H, and a scaling relation

is obtained with P = 1.4. Such a relation, which in- volves Hstop, indicates the presence of non-local rheo- logical properties for a dense granular flow. It is likely then, that the force chains dynamics, controls the transfer of momentum down to the base of the flow. A model establishing a link with the rheology of dense suspensions was proposed along those lines [45°00]. At larger angles, the flow accelerates and a theoretical frame work for thin-flows was developed by Savage et al. [46], which relies on the empirical existence of a dynamical angle of friction controlling the rheology. A high shearing rate, another regime of steady state flow may be reached when the agitation is such that the flow can be described by a series of binary collisions and behaves like a dissipative gas [4,45]. Then, the local shear stress T scales like:

where, p is the density, d a grain size, VV is a local velocity gradient and f ( v ) is a function diverging at large compacities v. A complex behavior of non-linear density waves is then predicted for vertical chutes [47]. In real systems, the collisional regime was observed either in the most diluted parts of the flows or for model systems with low dissipation on collisions [ 39 O*O].

298 E. Clement /Current Opinion in Colloid & Interface Science 4 (1999) 294-299

6. Conclusion

The most recent contributions to the rheology of granular flows were proposed to provide evidence or to account for many phenomenon dealing with the central problem of arch formation. This aspect is encountered either in static pilings or in quasi-static motion of strained assemblies and in dense granular flows such as avalanches. It is clear that due to this effect the two classical paradigms for granular rheology, such as the solid-on-solid friction pheno- menology and the theory of dissipative gases come short to explain what is currently observed (stress fluctuations, shear band structures, velocity selection, dynamic arches, etc.). At the moment, there is no clear agreement on how this difficult issue should be treated and many different visions are opposed. It is possible that the puzzle may be solved by crucial experiments and by a deeper understanding of the relevant dynamical aspects of force chains using a general frame-work called the ‘fabric tensors’ approach [48,491. This way establishes a rigorous (but so far useless in practice) link between the microscopic description of contact forces and the macroscopic world of stresses and strains. Furthermore, it is not clear, that the usual assumptions, which are made to describe contact force for non-cohesive granulates, are always sufficient to capture all the relevant macroscopic aspects. Many important features such as sensitivity to humidity, electrostatic effects, plastic yield on contact may lead to phenomenon such as aging or vault formation - effects which are directly observable for finer powders [50].

References and recommended reading

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