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 present an experiment which aim is to investigate the mechanical properties of a static granular assembly. The piling is an horizontal 3D granular layer confined in a box, we apply a localized extra force at the surface and the spatial distribution of stresses at the bottom is obtained (the mechanical Green's function). For different types of granular media, we observe a linear pressure response which profile shows one peak centered at the vertical of the point of application. The peak's width increases linearly when increasing the depth. This green function seems to be in -at leastqualitative agreement with predictions of elastic theory.

We measured the vertical pressure response function of a layer of sand submitted to a localized normal force at its surface. We found that this response profile depends on the way the layer has been prepared: all profiles show a single centered peak whose width scales with the thickness of the layer, but a dense packing gives a wider peak than a loose one. We calculate the prediction of isotropic elastic theory in the presence of a bottom boundary and compare it to the data. We found that the theory gives the right scaling and the correct qualitative shape, but fails to really fit the data.