Granular packings: Nonlinear elasticity, sound propagation, and collective relaxation dynamics

Abstract

Experiments on isotropic compression of a granular assembly of spheres show that the shear and bulk moduli vary with the confining pressure faster than the 1/3 power law predicted by HertzMindlin effective medium theories (EMT) of contact elasticity. Moreover, the ratio between the moduli is found to be larger than the prediction of the elastic theory by a constant value. The understanding of these discrepancies has been a longstanding question in the field of granular matter.Here we perform a test of the applicability of elasticity theory to granular materials. We perform sound propagation experiments, numerical simulations and theoretical studies to understand the elastic response of a deforming granular assembly of soft spheres under isotropic loading. Our results for the behavior of the elastic moduli of the system agree very well with experiments. We show that the elasticity partially describes the experimental and numerical results for a system under compressional loads. However, it drastically fails for systems under shear perturbations, particularly for packings without tangential forces and friction. Our work indicates that a correct treatment should include not only the purely elastic response but also collective relaxation mechanisms related to structural disorder and nonaffine motion of grains.

Goddard rattler-jamming mechanism for quantifying pressure dependence of elastic moduli of grain packs

“…Similarly, the upper limit corresponds to grains so rough that no slip occurs, resulting in the maximum transmitted shear at a contact. Acoustic data collected on unconsolidated sand and glass bead packs as presented in numerous investigations [1,2,[13][14][15][16][17] demonstrate that over the approximate stress range of 10 −3 to 10 MPa (corresponding to sediment overburden depths ranging from 10 −1 to 10 3 m), elastic moduli of the dry frame vary more rapidly than the Hertzian prediction of P 1/3 . At lower levels of applied pressure (<10 MPa), power-law pressuredependence is generally observed to have an exponent larger than the Hertzian 1/3.…”

Reflection of Nonlinear Acoustic Waves from the Mechanically Free Surface of an Unconsolidated Granular Medium

“…The experimental observations of higher powers up to α=1/4 (both for longitudinal and shear waves) could be theoretically attributed to different factors [33,34,18], such as increasing with pressure number of contacts between a bead and its neighbors, i.e. to an increasing coordination number [18].…”

“…Multiple experiments [14][15][16][17][18]24] demonstrate that in unconsolidated granular media the velocity of both longitudinal and shear acoustic waves is the power function of pressure (of normal stress) and, consequently, in accordance with equation (3), of depth coordinate theory [39,40] predicts α=1/6 both for longitudinal and shear acoustic waves [32,18]. The experimental observations of higher powers up to α=1/4 (both for longitudinal and shear waves) could be theoretically attributed to different factors [33,34,18], such as increasing with pressure number of contacts between a bead and its neighbors, i.e.…”

“…Both experiments [14][15][16][17][18] and theory [4-9, 12, 32-34] reveal power-law dependencies of elastic moduli on pressure and, consequently on depth. The peculiar feature of this gravity-induced stratification of a granular material is vanishing of the linear elastic moduli exactly at the free surface of a cohesionless granular material, where in the effective medium approximation the pressure equals zero, and the ratio of the linear modulus to the nonlinear modulus diverges.…”

“…In the currently existing terminology the most important "classical" acoustical experiments in granular medium, consisting in the measurements of the sound velocity dependences on pressure [11,[14][15][16][17][18], can be called acousto-elasticity experiments and are considered as nonlinear acoustical experiments [19,20].…”