Effects of friction and disorder on the quasistatic response of granular solids to a localized force.

The response to a localized force provides a sensitive test for models of stress transmission in granular solids. Elasto-plastic models, traditionally used by engineers, have been challenged by theories and experiments that suggest a wavelike (hyperbolic) propagation of the stress, as opposed to the elliptic equations of static elasticity. Simulations of two-dimensional granular systems subject to a localized external force have been employed to examine the nature of stress transmission in these systems as a function of the magnitude of this force, the frictional parameters, and degree of disorder. The results indicate that in large systems (as considered by engineers) the response is close to that predicted by isotropic elasticity, whereas for small systems (or strongly forced ones) it is strongly anisotropic. In the latter case the applied force induces changes in the contact network accompanied by frictional sliding and gives rise to hyperboliclike stress propagation. The larger the static friction, the more extended the range of forces for which the response is elastic, and the smaller the anisotropy. Increase in the degree of polydispersity (in the studied range, up to 25%) decreases the range of elastic response. This paper is an extension of a previously published Letter [C. Goldenberg and I. Goldhirsch, Nature (London) 435, 188 (2005)].

Scale separation in granular packings: stress plateaus and fluctuations.

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 enabling an "objective" definition of this field. Expectedly, it is not the mere size of 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 subparticle 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.

Friction enhances elasticity in granular solids.

For years, engineers have used elastic and plastic models to describe the properties of granular solids, such as sand piles and grains in silos. However, there are theoretical and experimental results that challenge this approach. Specifically, it has been claimed that stress in granular solids propagates in a manner described by wave-like (hyperbolic) equations, rather than the elliptic equations of static elasticity. Here we report numerical simulations of the response of a two-dimensional granular slab to an external load, revealing that both approaches are valid--albeit on different length scales. For small systems that can be considered mesoscopic on the scale of the grains, a hyperbolic-like, strongly anisotropic response is expected. However, in large systems (those typically considered by engineers), the response is closer to that predicted by traditional isotropic elasticity models. Static friction, often ignored in simple models, plays a key role: it increases the elastic range and renders the response more isotropic, even beyond this range.

Hydrodynamics of nearly smooth granular gases.

Hydrodynamic equations of motion for a monodisperse collection of nearly smooth homogeneous spheres have been derived from the corresponding Boltzmann equation, using a Chapman-Enskog expansion around the elastic smooth spheres limit. Because in the smooth limit the rotational degrees of freedom are uncoupled from the translational ones, it turns out that the required hydrodynamic fields include (in addition to the standard density, velocity, and translational granular temperature fields) the (infinite) set of number densities, n(s,r, t), corresponding to the continuum of values of the angular velocities. The Chapman-Enskog expansion was carried out to high (up to 10th) order in a Sonine polynomial expansion by using a novel computer-aided method. One of the consequences of these equations is that the asymptotic spin distribution in the homogeneous cooling state for nearly smooth, nearly elastic spheres, is highly non-Maxwellian. The simple sheared flow possesses a highly non-Maxwellian distribution as well. In the case of wall-bounded shear, it is shown that the angular velocity injected at the boundaries has a finite penetration length.

Force chains, microelasticity, and macroelasticity.

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 departures 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.

On the microscopic foundations of elasticity.

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 based 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.

Scales and kinetics of granular flows.

When a granular material experiences strong forcing, as may be the case, e.g., for coal or gravel flowing down a chute or snow (or rocks) avalanching down a mountain slope, the individual grains interact by nearly instantaneous collisions, much like in the classical model of a gas. The dissipative nature of the particle collisions renders this analogy incomplete and is the source of a number of phenomena which are peculiar to "granular gases," such as clustering and collapse. In addition, the inelasticity of the collisions is the reason that granular gases, unlike atomic ones, lack temporal and spatial scale separation, a fact manifested by macroscopic mean free paths, scale dependent stresses, "macroscopic measurability" of "microscopic fluctuations" and observability of the effects of the Burnett and super-Burnett "corrections." The latter features may also exist in atomic fluids but they are observable there only under extreme conditions. Clustering, collapse and a kinetic theory for rapid flows of dilute granular systems, including a derivation of boundary conditions, are described alongside the mesoscopic properties of these systems with emphasis on the effects, theoretical conclusions and restrictions imposed by the lack of scale separation. (c) 1999 American Institute of Physics.