densities of both phases, which are depicted by the plain (dashed) red curves in Fig. 3(b) .
To conclude, we investigated the dynamics of large tracer particles placed in a mechanically excited **granular** **gas**. System density plays a crucial role in the observed dynamics. Indeed, above some threshold filling, tracers experience successive episodes of local trapping yielding in large mobility fluctuations. Particle tracking has shown that the velocity distributions of the tracers are strongly linked to the state of the system and can be used to infer whether a liquid phase coexists within a **granular** **gas**. Finally, we developed a theoretical model based on Onsager ’s mini- mum rate of dissipation principle, that fits our observations. We thank M. Braibanti and V. Koehne from ESA as well as R. Stannarius from Otto von Guericke University for fruitful discussions. VIP-Gran-PF instrument was built by DTM Technologies (Modena, Italy). This work was funded by European Space Agency Topical Team SpaceGrains No. 4000103461. We thank the support of Novespace during ESA Parabolic Flight Campaigns. M. N. thanks the Belgian Federal Science Policy Office (BELSPO) for the provision of financial support in the framework of the PRODEX Programme of the European Space Agency (ESA) under Contract No. 4000103267. E. F., Y. G., C. L.-C., and F. P. thank CNES for partial financial support. D. F. acknowledges funding by DLR within project EQUIPAGE II (50WM1842).

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[9] E. F ERRARI , L. P ARESCHI , Modelling and numerical methods for the diffusion of impurities in a **gas**, Int. J. Numer. Meth. Fluids, to appear.
[10] V. G ARZ O ` , Kinetic Theory for Binary **Granular** Mixtures at Low-Density, preprint, ArXiv:0704.1211 . [11] V. G ARZ O ` , J. M. M ONTANERO , Diffusion of impurities in a **granular** **gas**, Physical Review E, 69, 2004. [12] H. G RAD , Asymptotic theory of the Boltzmann equation. II. Rarefied **Gas** Dynamics (Proc. 3rd Internat. Sympos.,

is increased, the competition between repulsive interac- tions and kinetic agitation results in a transition from a **granular** **gas** towards a hexagonal crystal. Snapshots of a window inside S are shown without applied magnetic field in Fig. 1 (b) (ε ≈ 0) and with a moderate value of B = 62 G (ε ≈ 8.80) in Fig. 1 (c). In both cases, the assembly of spheres is in a **granular** **gas** state, but in the second snapshot the particles do not come into contact anymore. We note also a smaller number of particles in the second case. In Fig. 2 (b), we show indeed a de- crease of φ with ε in the observation window S, which is due to increasing particle repulsion while boundaries are non-repulsive. The crystallization towards a hexago- nal crystal is monitored by the sixfold bond-orientational order parameter per particle

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V. CONCLUSIONS
In this paper, we have presented strong hints that the functional given by Eq. (42) can play the role of a Lyapunov functional in the context of a dissipative **granular** **gas**: it decays monotonically in time, tending to zero in the late time non-equilibrium steady state. These results, in agreement with those of Ref [22], have been shown by three different kinds of simulation methods, for a wide class of initial conditions and a wide range of inelasticities, 0 < α < 1. Our functional –that takes the form of a relative entropy, or Kullback-Leibler distance– reduces to Boltzmann’s original H in the case of elastic interactions. Its very form can be directly inspired from information theory, where the Kullback-Leibler distance plays a prominent role [39].

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the experimental ones, and none of them predict even the most basic features of the experimental results. For example, numerical and theoretical studies often assume that the vibration frequency is very high so that the shaking wall can be replaced by a thermal boundary. The vibrated **granular** **gas** thus attains a steady state. But very high frequencies with an amplitude sufficient to fluidize the **granular** materials are difficult to attain experimentally; the vibration amplitude in experiments is often a significant fraction of the size of the container. In experiments, the pressure and **granular** temperature all vary strongly over the period of one vibration cycle. It has therefore been im- possible to compare the experimental and numerical results in a meaningful way.

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is totally inelastic. This completely inelastic bounce was observed
when m i ? m c for different filling volumes V f (see Fig. 5a). Only the sphere completely full of grains and the empty sphere bounce several times before a complete stop. For partially filled spheres the coefficient of restitution is always zero at the second impact, see Fig. 5b. It is important to clarify that the damping of the first rebound increases with V f due to the larger deformation of the ball, as was corroborated using spheres with equivalent mass of liquid. Nevertheless, following the second impact, the liquid-filled sphere displays a series of small consecutive bounces, whereas the grain- filled sphere stops dramatically, see Figs. 5c–d. Snapshots in Fig. 6 reveal that the liquid reacts more quickly to the sphere impact than the grains. However, whereas the liquid is always in contact with the sphere surface, the grains are decoupled from the sphere and form a **granular** **gas** that collapses at the second impact. This abrupt collapse dissipates all the energy of the system. In contrast, the water con- tinues splashing inside the other sphere which remains rebounding. Additional research in this regard and the role of confinement pro- duced by the geometry will be presented in a future work.

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that the shear strength is mainly due to the increasing aptitude of
the particles to self-organize in an anisotropic network 41 , 44 .
Discussion
The additive rheology of **granular** materials, as demonstrated in this paper, is by no means self-evident. The expectation that a system involving several dimensionless parameters can ultimately be described by a single parameter combining those parameters is unusual. The deep reason behind such a behavior is the very nature of **granular** materials in which the particle interactions are concentrated at the contact points and the local dynamics is controlled by the shear rate. Hence, by careful distinction of stresses depending on the shear rate (ﬁrst group) from those that are independent (second group), a single parameter can be deﬁned by means of the stress additivity property. In this respect,

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Understanding how **granular** materials self-organize into mechanically stable states requires consideration of the structure of force networks. Figure 1 shows exam- ples of typical force networks found in the experiments and simulations. A full description of the force network requires a high dimensional space that reports on micro- scopic features and does not directly reveal its mesoscopic structure. In order to understand this structure, we need statistical tools for their characterization that are sensi- tive, systematic, unbiased, minimal, and consistent with macroscopic properties, such as the system-wide stresses. These tools must also be able to distinguish different states of the system, based on force networks.

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scale can be defined. This last point may be responsible for the selection of a velocity gradient independent of both
R and u, i.e., independent of the shear stress.
The observation of the successive clusterized frames shows that the largest clusters are emitted by the static phase and die either by fragmenting into smaller ones or by sticking back to the static phase. Differences between **granular** flows down a rough inclined plane and **granular** surface flows can then be rationalized as follows: cluster- ing results from the competition between inelastic multiple collisions, which tends to aggregate grains together [11], and shear, which erodes clusters. For flows down an in- clined plane, these two effects lead to clusters of a typical size, whereas in surface flows, the static bed plays the role of a cluster reservoir: its erosion by the flowing grains can generate very large size clusters that then split and cas- cade into smaller and smaller ones (this mechanism might explain why we observe power-law distributions). In this view, the cluster size distribution, and a fortiori the veloc- ity gradient, depend crucially on the boundary conditions. The absence of a characteristic correlation length to de- scribe the locally “jammed” clusters of beads is quite in- teresting. First, this contrasts with an assumption made in most local and nonlocal models [4,6,8,10], where the tran- sition between the “solid” phase and the “liquid” phase is supposed to occur over a well defined length scale. The ex- istence of multiscale rigid clusters indicates that the flow- ing phase is actually critical and suggests the proximity of a continuous “jamming” transition, of the type proposed in [17] (see also [18] for a related discussion). In this respect, it is interesting to note that similar power-law distributed clusters of strongly correlated motion have recently been observed in a colloid close to the glass transition [19].

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CEMAGREF – Unite´ ETNA – Grenoble, France; d 3S-R Laboratory – INPG, UJF, CNRS – Grenoble, France
We study the evolution of structure inside a deforming, cohesionless **granular** material undergoing failure in the absence of strain localisation –so-called diffuse failure. The spatio-temporal evolution of the basic building blocks for self-organisation (i.e. force chains and minimal contact cycles) reveals direct insights into the structural origins of failure. Irrespective of failure mode, self-organisation is governed by the cooper-ative behaviour of truss-like 3-cycles providing lateral support to column-like force chains. The 3-cycles, which are initially in scarce supply, form a minority subset of the minimal contact cycle bases. At large length-scales (i.e. sample size), these structures are randomly dispersed, and remain as such while their population progressively falls as loading proceeds. Bereft of redundant constraints from the 3-cycles, the force chains are initially just above the isostatic state, a condition that progressively worsens as the sample dilates. This diminishing capacity for redistribution of forces without incurring physical rearrangements of member particles renders the force chains highly prone to buckling. A multiscale analysis of the spatial patterns of force chain buckling reveals no clustering or localisation with respect to the macroscopic scale. Temporal patterns of birth-and-death of 3-cycles and 3-force chains provide unambiguous evidence that significant structural reorganisations among these building blocks drive rheological behaviour at all stages of the loading history. The near-total collapse of all structural building blocks and the spatially random distribution of force chain buckling and 3-cycles hint at a possible signature of diffuse failure.

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1. Introduction
When a non-cohesive **granular** media (like dry sand) is poured from a single point, it forms a conical pile. The angle of the free surface from the horizontal / is called the angle of repose ( Fig. 1 -a). The explanation, given by Coulomb, is that **granular** media sustain shear stresses as in solid friction. The pile remains stable if in all parts we have s < lr, where s and r are, respectively, the shear and normal stress; l is called the apparent coefficient of friction of the **granular** media.

Keywords: **Granular** materials; grain breakage; elasto-plasticity; critical state; grain size distribution.
1. INTRODUCTION
Particle crushing occurred along both compression and shearing stress paths, especially under high confining stress (e.g. within earth dams, deep well shafts). This phenomenon is however more intense during shearing. The influence of particle crushing on the mechanical behavior of **granular** materials has been widely investigated in the past decades.

protocols implemented are described in the following.
2.3.1 **Granular** rheometer
FT4 rotational rheometer (Freeman Technology, UK) was used to carry out the experimental tests in shear measurements. Measurements were carried out in a cylindrical glass cell of 25 mm in diameter and 10 mm in height (Figure 2-a). The cell was filled to capacity with sludge, as shown in Figure 2-b. To create a uniform packing throughout the sample, remove any variability during the preparation and thus improve results reproducibility, a pre- consolidation stress σ c (without any rotation) was applied for 60 s (Figure 2-c) before every test.

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mobilization increases until the grains may move down- wards. Janssen ’s profile can be recovered with great accu- racy provided a large amount of **granular** material is released during the discharge [38,42] . In our experiments, we do not expect a full friction mobilization after pouring and collapse occurs at the discharge onset. Consequently, to model, in the simplest way, the fact that the upward friction polarization at the wall may not be achieved when the collapse occurs, we introduce an empirical dimensionless parameter ξ such that λ ¼ ξR, where ξ can be varied from ξ ¼ Oð1Þ (full mobilization) to ξ ≫ L=R (random mobilization).

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5 Conclusion
Our extensive simulations using a sub-particle well- resolved computation of a viscous ﬂuid coupled to rigid particles suggest that the visco-inertial ﬂow regime of im- mersed **granular** materials can be described by a modiﬁed inertial number combining the inertial number with the Stokes number. This behavior can also be expressed by using a viscous description of the ﬂow. In this description, both shear viscosity and bulk viscosity can be described by unique function of the packing fraction provided the ﬂuid viscosity η f is replaced by ηf + ρs d 2 ˙γ/α. This remarkable scaling reﬂects the joint eﬀects of particle inertia and ﬂuid viscous forces on the **granular** microstructure. A detailed analysis of local evolutions of pores pressures and contact networks will be presented elsewhere.

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PDMS coatings on asymmetric **gas** separation membranes: effect on pure **gas** and mixed **gas** selectivity

Idealised test cases showed that the present numerical scheme correctly handled the viscous-plastic regime. A solution of the full two-phase flow problem with realistic geometry and boundary conditions was then developed. The results show that the frictional properties of the solid phase play an important, previously neglected role. The pressure distribution down the vessel is quite different from that obtained by neglecting friction. The present model indicated a relatively constant pressure along the vessel walls; a result which is in agreement with numerous observations of **granular** materials. The pressure distribution, and indeed the motion of the solid phase also appeared to be sensitive to the frictional properties of the **granular** material in the vicinity of fluid inlets, a situation with important ramifications to the equipment designer and operator.

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Study of drainage (schema 3 in Fig. 2 )
Then the foam tube is turned to the vertical and we start to measure the drainage kinetics of the sample. We follow the evolution of the height ℎ(𝑡) locating the transition between the foam and the drained suspension at the bottom of the column (see Fig. 3 ). Note that the pictures such than those presented in Fig. 3 were used to check for the absence of coalescence during the measurements. As already pointed out [18,21] the drainage behavior of foamy suspensions exhibits a linear regime, characterized by a well-defined and constant drainage velocity, provided that 𝑡 ≲ 𝜏, where 𝜏 is the half drainage time. During this regime, the volume of liquid/suspension drained out of the foam has flowed through foam areas that have not yet been reached by the drainage front, i.e. areas where the **gas** fraction has remained equal to the initial value 𝜙. Because the linear regime accounts for drainage properties of foam characterized by a constant **gas** fraction 𝜙, we measure the drainage velocity 𝑉 from the slope of this linear evolution, 𝑉 = 𝑑ℎ 𝑑𝑡 ⁄ . Note that of all drainage curves were found to exhibit the linear regime, as shown in Fig. 3 . In order to characterize the effect of particles on drainage, we normalize the measured drainage velocity by the one measured without particle, i.e. 𝑉̃ ≡ V(𝜑 𝑝 ) V(0) ⁄ . Note that because of uncertainties related to the measurement of ℎ(𝑡) for ℎ ≃ 0,

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This equation preserves mass and positivity and we shall be concerned with solutions which
are probability measures on R n at all times. It is used in the modelling of space-homogeneous
**granular** media (see [3]), where it governs the evolution of the velocity distribution µ t (x) of a
particle under the effects of diffusion, a possible exterior potential V and a mean field interaction through the potential W ; we shall keep the variable x instead of v (for the velocities) for notational convenience.

inertial number, which can also be interpreted as a generalized fluidity parameter of the suspension tending to the inverse
of effective viscosity in the viscous limit. The robustness of the visco-inertial number was shown for frictional and viscous descriptions of the flow as well as for the **granular** microstructure by varying several system parameters, includ- ing relative fluid-grain density, in a broad range of values. The rheology is mainly governed by the contact network anisotropy reflecting the joint effects of viscous and inertial stresses. Our results obtained by means of subparticle computational fluid dynamics simulations considerably extend the scope of a unique framework introduced by Trulsson et al. [ 6 ] to a more general parameter space and to the descriptors of microstructural anisotropy. It opens the way also to detailed analysis of local evolutions of pore pressures and contact networks. Despite computational effort, the extension of this investigation to 3D and turbulent flows is desirable.

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