We experimentally determine ensemble-averaged responses of granular packings to point forces, and we compare these results to recent models for force propagation in a granular material. We used 2D granular arrays consisting of photoelastic particles: either disks or pentagons, thus spanning the range from ordered to disordered packings. A key finding is that spatial ordering of the particles is a key factor in the force response. Ordered packings have a propagative component that does not occur in disordered packings.PACS numbers: 46.10.+z, 47.20.-k Granular systems have captured much recent interest due to their rich phenomenology, and important applications [1]. Even in the absence of strong spatial disorder of the grains, static arrays show inhomogeneous spatial stress profiles called stress (or force) chains [2]. Forces are carried primarily by a tenuous network that is a fraction of the total number of grains.A fundamental unresolved issue concerns how granular materials respond to applied forces, and there are several substantially different models. A broad group of conventional continuum models (e.g. elasto-plastic, . . .) posit an elastic response for material up to the point of plastic deformation [3]. The stresses in portions of such a system below plastic yield have an elastic response and satisfy an elliptic partial differential equation (PDE); those parts that are plastically deforming satisfy a hyperbolic PDE. Several fundamentally different models have recently been proposed. The q-model of Coppersmith et al.[4] assumes a regular lattice of grains, and randomness is introduced at the contacts. This model successfully predicts the distribution of forces in the large force limit, as verified by several static and quasistatic experiments and models [4][5][6]. In the continuum limit, this model reduces to the diffusion equation, since the forces effectively propagate by a random walk. Another model (the Oriented Stress Linearity-OSL-model) of Bouchaud et al. [7], has a constitutive law, justified through a microscopic model, of the form σ zz = µσ xz + ησ xx (in 2D) in order to close the stress balance conditions ∂σ ij /∂x j = ρg i . This leads to wave-like hyperbolic PDEs describing the spatial variation of stresses. In later work, these authors considered weak randomness in the lattice The range of predictions among the models is perhaps best appreciated by noting that the different pictures predict qualitatively different PDEs for the variation of stresses within a sample: e.g. for elasto-plastic models an elliptic or hyperbolic PDE; for the q-model, a parabolic PDE; and for the OSL model without randomness, a hyperbolic PDE. The impact of equation type extends to the boundary conditions needed to determine a solution: e.g. hyperbolic equations require less boundary information than an elliptic equation.Here, we explore these issues through experiments on a 2D granular system consisting of photoelastic (i.e birefringent under strain) polymer particles [6] that are either disks or pentagons. By vi...

We describe experiments that probe the response to a point force of 2D granular systems under a variety of conditions. Using photoelastic particles to determine forces at the grain scale, we obtain ensembles of responses for the following particle types, packing geometries and conditions: monodisperse ordered hexagonal packings of disks, bidisperse packings of disks with different amount of disorder, disks packed in a regular rectangular lattice with different frictional properties, packings of pentagonal particles, systems with forces applied at an arbitrary angle at the surface, and systems prepared with shear deformation, hence with texture or anisotropy. We experimentally show that disorder, packing structure, friction and texture significantly affect the average force response in granular systems. For packings with weak disorder, the mean forces propagate primarily along lattice directions. The width of the response along these preferred directions grows with depth, increasingly so as the disorder of the system grows. Also, as the disorder increases, the two propagation directions of the mean force merge into a single direction. The response function for the mean force in the most strongly disordered system is quantitatively consistent with an elastic description for forces applied nearly normally to a surface, but this description is not as good for non-normal applied forces. These observations are consistent with recent predictions of Bouchaud et al. [Bouchaud et al., Euro. Phys. J. E4 451 (2001); Socolar et al., Euro. Phys. J. E7 353 (2002)] and with the anisotropic elasticity models of Goldenberg and Goldhirsch [Goldenberg & Goldhirsch, Phys. Rev. Lett. 89 084302 (2002)]. At this time, it is not possible to distinguish between these two models. The data do not support a diffusive picture, as in the q-model, and they are in conflict with data by Rajchenbach [Da Silva & Rajchenbach, Nature 406 708 (2000)] that indicate a parabolic response for a system consisting of cuboidal blocks. We also explore the spatial properties of force chains in an anisotropic textured system created by a nearly uniform shear. This system is characterized by stress chains that are strongly oriented along an angle of 45 o , corresponding to the compressive direction of the shear deformation. In this case, the spatial correlation function for force has a range of only one particle size in the direction transverse to the chains, and varies as a power law in the direction Preprint submitted to Elsevier Science of the chains, with an exponent of -0.81. The response to forces is strongest along the direction of the force chains, as expected. Forces applied in other directions are effectively refocused towards the strong force chain direction.

We study the drag force experienced by an object slowly moving at constant velocity through a two-dimensional granular material consisting of bidisperse disks. The drag force is dominated by force chain structures in the bulk of the system, thus showing strong fluctuations. We consider the effect of three important control parameters for the system: the packing fraction, the drag velocity and the size of the tracer particle. We find that the mean drag force increases as a power law (exponent of 1.5) in the reduced packing fraction, (gamma- gamma(c) ) / gamma(c) , as gamma passes through a critical packing fraction, gamma(c) . By comparison, the mean drag grows slowly (basically logarithmic) with the drag velocity, showing a weak rate dependence. We also find that the mean drag force depends nonlinearly on the diameter, a of the tracer particle when a is comparable to the surrounding particles' size. However, the system nevertheless exhibits strong statistical invariance in the sense that many physical quantities collapse onto a single curve under appropriate scaling: force distributions P (f) collapse with appropriate scaling by the mean force, the power spectra P (omega) collapse when scaled by the drag velocity, and the avalanche size and duration distributions collapse when scaled by the mean avalanche size and duration. We also show that the system can be understood using simple failure models, which reproduce many experimental observations. These observations include the following: a power law variation of the spectrum with frequency characterized by an exponent alpha=-2 , exponential distributions for both the avalanche size and duration, and an exponential fall-off at large forces for the force distributions. These experimental data and simulations indicate that fluctuations in the drag force seem to be associated with the force chain formation and breaking in the system. Moreover, our simulations suggest that the logarithmic increase of the mean drag force with rate can be accounted for if slow relaxation of the force chain networks is included.

The measurement of force distributions in sandpiles provides a useful way to test concepts and models of the way forces propagate within noncohesive granular materials. Recent theory [J.-P. Bouchaud, M.E. Cates, and P. Claudin, J. Phys. I 5, 639 (1995); M. E. Cates, J. P. Wittmer, J.-P. Bouchaud, and P. Claudin, Phil. Trans. Roy. Soc. 356, 2535 (1998)] by Bouchaud et al. implies that the internal structure of a heap (and therefore the force pathway) is a strong function of the construction history. In general, it is difficult to obtain information that could test this idea from three-dimensional granular experiments except at boundaries. However, two-dimensional systems, such as those used here, can yield information on forces and particle arrangements in the interior of a sample. We obtain position and force information through the use of photoelastic particles. These experiments show that the history of the heap formation has a dramatic effect on the arrangement of particles (texture) and a weaker but clear effect on the forces within the sample. Specifically, heaps prepared by pouring from a point source show strong anisotropy in the contact angle distribution. Depending on additional details, they show a stress dip near the center. Heaps formed from a broad source show relatively little contact angle anisotropy and no indication of a stress dip.

Abstract. We relate the pressure 'dip' observed at the bottom of a sandpile prepared by successive avalanches to the stress profile obtained on sheared granular layers in response to a localized vertical overload. We show that, within a simple anisotropic elastic analysis, the skewness and the tilt of the response profile caused by shearing provide a qualitative agreement with the sandpile dip effect. We conclude that the texture anisotropy produced by the avalanches is in essence similar to that induced by a simple shearing -albeit tilted by the angle of repose of the pile. This work also shows that this response function technique could be very well adapted to probe the texture of static granular packing. The stress distribution below a pile of sand has been one of the problematic issues of the statics of granular materials in physics over the last few years [1]. In fact, experiments have shown that, when a granular pile is prepared from a point source, the bottom pressure profile has a clear local minimum -a 'dip' -below the apex [2,3,4]. The existence of this pressure dip has been strongly debated, and it is now well established that the presence or absence of this dip is closely related to the preparation history of the pile. This was demonstrated by Vanel et al. [4]. Using the same sand and experimental apparatus, these authors could generate the stress dip using a localized deposition technique or cause the dip to disappear by constructing a pile in successive horizontal layers. Similar conclusions were reached for two dimensional heaps with photo-elastic grains [5], and in numerical simulations [6,7,8].This interesting effect has inspired the development of new models to describe how forces are transmitted in dense granular materials. Among them are those proposed by Bouchaud et al., initially developed in the context of the sand pile dip [9,10,11,12], and further extended to other geometries like that of the silo [12,13]. This approach is also intended to describe a collection of systems including dense colloids, granular matter or foams [16]. At the macroscopic level, these features are modelled by hyperbolic, partial differential equations (PDE) for the stress tensor. Although no explicit link was established, the characteristics of these hyperbolic equations were intuitively thought to be related to the mesoscopic 'force chain' network whose structure and orientation were shaped by the previous history of the granular assembly -see also [14,15] concerning force chains. Plasticity theories for granular deformations are also of hyperbolic type, although conceptually different than the previous cited models. From the classical, soil mechanics point of view, below the plastic threshold, granular material is thought to behave as an effective elastic material with PDE's that belong to the elliptic class [17]. Finally, sound wave propagation techniques and numerical simulations of confined granular assemblies indicate that that assesment of effective elastic constitutive relations is still an open and diff...