Footprints in Sand: The Response of a Granular Material to Local Perturbations

Abstract

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...

Photoelastic force measurements in granular materials

“…Even in modern experiments, there is much that can be learned from comparing the spatial patterns of the force chains. For example, images of the force chains can reveal history dependence via spatial correlations [34,35], the ensemble-averaged response to point forces [36], and the principal axis of the response [37]. By subtracting images taken before/after an event, it is possible to examine spatial patterns of failure [19,38] and the speed of sound [24].…”

Logarithmic rate dependence of force networks in sheared granular materials

“…By using a photoelastic material, we could determine forces at the grain scale. More detailed descriptions of the experimental apparatus and methods [21] have been given elsewhere for related experiments [18,22,23], and here we provide only essential information for two qualitatively different experiments. For each experiment, the humidity varied by no more than ±5%.…”

Monitoring three-dimensional packings in microgravity

“…While for three dimensions, granular stress-birefringence has so far remained largely on the qualitative level, in two dimensions, granular stress-birefringence (also called photoelasticity) has been utilized in a large variety of instances and analyzed in great detail: In sheared systems the transition between loose and load-carrying packings was established [17]; a qualitative difference in force response to external loads was found for ordered and disordered packings [18]; logarithmic aging of stress was discovered for packings but not observed under compression [19]; the relation between force chains and friction in stick-slip motion was elaborated [20]; for granular sound, the propagation along force chains was demonstrated [21]; for jamming under isotropic compression, non-trivial power-laws were confirmed [22,23,24] and distinguished from jamming behavior under shear [25,26,27,28].…”