The response to a localized force provides a sensitive test for different models of stress transmission in granular solids. The elasto-plastic models traditionally used by engineers have been challenged by theoretical and experimental results which s
uggest a wave-like (hyperbolic) propagation of the stress, as opposed to the elliptic equations of static elasticity. Numerical simulations of two-dimensional granular systems subject to a localized external force are employed to examine the nature of stress transmission in these systems as a function of the magnitude of the applied force, the frictional parameters and the disorder (polydispersity). The results indicate that in large systems (typically considered by engineers), the response is close to that predicted by isotropic elasticity whereas the response of small systems (or when sufficiently large forces are applied) is strongly anisotropic. In the latter case the applied force induces changes in the contact network accompanied by frictional sliding. The larger the coefficient of static friction, the more extended is the range of forces for which the response is elastic and the smaller the anisotropy. Increasing the degree of polydispersity (for the range studied, up to 25%) decreases the range of elastic response. This article is an extension of a previously published letter [1].
One goal of this paper is to discuss the classical definition of granular temperature as an extension of its thermodynamic equivalent and a useful concept which provides an important characterization of fluidized granular matter. Following a review o
f some basic concepts and techniques (with emphasis on fundamental issues) we present new results for a system that can exhibit strong violations of equipartition, yet is amenable to description by classical granular hydrodynamics, namely a binary granular gas mixture. A second goal of this article is to present a result that pertains to dense granular and molecular solids alike, namely the existence of a correction to the elastic energy which is related to the heat flux in the equations of continuum mechanics. The latter is of the same (second) order in the strain as the elastic energy. Although recent definitions of temperatures for granular matter, glasses and other disordered many-body systems are not within the scope of this article we do make several general comments on this subject in the closing section.
We study the local disorder in the deformation of amorphous materials by decomposing the particle displacements into a continuous, inhomogeneous field and the corresponding fluctuations. We compare these fields to the commonly used non-affine displac
ements in an elastically deformed 2D Lennard-Jones glass. Unlike the non-affine field, the fluctuations are very localized, and exhibit a much smaller (and system size independent) correlation length, on the order of a particle diameter, supporting the applicability of the notion of local defects to such materials. We propose a scalar noise field to characterize the fluctuations, as an additional field for extended continuum models, e.g., to describe the localized irreversible events observed during plastic deformation.
We measure experimentally the rearrangements due to a small localized cyclic displacement applied to a packing of rigid grains under gravity in a 2D geometry. We analyze the evolution of the response to this perturbation by considering the individual
particle displacement and the coarse grained displacement field, as well as the mean packing fraction and coordination number. We find that the displacement response is rather long ranged, and evolves considerably with the number of cycles. We show that a small difference in the preparation method (induced by tapping the container) leads to a significant modification in the response though the packing fraction changes are minute. Not only the initial response but also its further evolution change with preparation, demonstrating that the system still retains a memory of the initial preparation after many cycles. Nevertheless, after a sufficient number of cycles, the displacement response for both preparation methods converges to a nearly radial field with a 1/r decay from the perturbation source. The observed differences between the preparation methods seem to be related to the changes in the coordination number (which is more sensitive to the evolution of the packing than the packing fraction). Specifically, it may be understood as an effect of the breaking of local arches, which affects the lateral transmission of forces.
It is demonstrated, by numerical simulations of a 2D assembly of polydisperse disks, that there exists a range (plateau) of coarse graining scales for which the stress tensor field in a granular solid is nearly resolution independent, thereby enablin
g an `objective definition of this field. Expectedly, it is not the mere size of the the system but the (related) magnitudes of the gradients that determine the widths of the plateaus. Ensemble averaging (even over `small ensembles) extends the widths of the plateaus to sub-particle scales. The fluctuations within the ensemble are studied as well. Both the response to homogeneous forcing and to an external compressive localized load (and gravity) are studied. Implications to small solid systems and constitutive relations are briefly discussed.
The modeling of the elastic properties of disordered or nanoscale solids requires the foundations of the theory of elasticity to be revisited, as one explores scales at which this theory may no longer hold. The only cases for which microscopically ba
sed derivations of elasticity are documented are (nearly) uniformly strained lattices. A microscopic approach to elasticity is proposed. As a first step, microscopically exact expressions for the displacement, strain and stress fields are derived. Conditions under which linear elastic constitutive relations hold are studied theoretically and numerically. It turns out that standard continuum elasticity is not self-evident, and applies only above certain spatial scales, which depend on details of the considered system and boundary conditions. Possible relevance to granular materials is briefly discussed.
The modeling of the elastic properties of granular or nanoscale systems requires the foundations of the theory of elasticity to be revisited, as one explores scales at which this theory may no longer hold. The only cases for which a microscopic justi
fication of elasticity exists are (nearly) uniformly strained lattices. A microscopic theory of elasticity, as well as simulations, reveal that standard continuum elasticity applies only at sufficiently large scales (typically 100 particle diameters). Interestingly, force chains, which have been observed in experiments on granular systems, and attributed to non-elastic effects, are shown to exist in systems composed of harmonically interacting constituents. The corresponding stress field, which is a continuum mechanical (averaged) entity, exhibits no chain structures even at near-microscopic resolutions, but it does reflect macroscopic anisotropy, when present.
It has been claimed that quasistatic granular materials, as well as nanoscale materials, exhibit departures from elasticity even at small loadings. It is demonstrated, using 2D and 3D models with interparticle harmonic interactions, that such departu
res are expected at small scales [below O(100) particle diameters], at which continuum elasticity is invalid, and vanish at large scales. The models exhibit force chains on small scales, and force and stress distributions which agree with experimental findings. Effects of anisotropy, disorder and boundary conditions are discussed as well.
