Nonlinear elasticity and pressure-dependent wave speeds in granular media

Abstract

Following is an analysis of the small-strain nonlinear elasticity of granular media near states of zero stress, as it relates to the pressure-dependent incremental linear elasticity and wave speeds. The main object is elucidation of the p ½ dependence of incremental elastic moduli on pressure p , a dependence observed in numerous experiments but found to be at odds with the p ½ scaling predicted by various micromechanical models based on hertzian contact. After presenting a power-law continuum model for small-strain nonlinear elasticity, the present work develops micromechanical models based on two alternative mechanisms for the anomalous pressure scaling, namely: (1) departures at the single-contact level from the hertzian contact, due to point-like or conical asphericity; (2) variation in the number density of hertzian contacts, due to buckling of particle chains. Both mechanisms result in p ½ pressure scaling at low pressure and both exhibit a high-pressure transition to p ½ scaling at a characteristic transition pressure p *. For assemblages of nearly equal spheres, a non-hertzian contact model for mechanism (1) and percolation-type model for (2) yield estimates of p * of the form p * = c μ ˆ ∝ 3 . Here c is a non-dimensional coefficient depending only on granular-contact geometry, while α ≪ 1 is a small parameter representing spherical imperfections and μ ˆ is an appropriate elastic modulus of the particles. Then, with R representing particle radius and h a characteristic spherical tolerance or asperity height, it is found that α = ( h / R ) ½ for mechanism (1) as opposed to α = h / R for (2). Limited data from the classic experiments of Duffy & Mindlin on sphere assemblages tend to support mechanism (1), but more exhaustive experiments are called for. In addition to the above analysis of reversible elastic effects, a percolation model of inelastic ‘shake-down’ or consolidation is given. It serves to describe how prolonged mechanical vibration, leading to the replacement of point-like or inactive contacts by stiffer Hertz contacts may change the pressure-scaling behaviour of particulate media. The present analysis suggests that pressure-dependence of elasticity may provide a useful means of characterizing the state of consolidation and stability of dense particulate media.

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

“…Although Goddard's ideas [10] have certainly received some attention [9,14,[17][18][19][20][21] in the granular packing literature, the present authors feel more attention to the details is appropriate, and this is the main contribution that is being made here. Goddard [10] was motivated to demonstrate that the elastic moduli increase with pressure as P 1/2 , instead of P 1/3 (which is again the result due to Hertz). Our goal is to explain the range of experimentally observed exponents on the pressure power law which will require the introduction of several new ideas and assumptions not made specifically by Goddard [10].…”

“…To account for the observed non-Hertzian pressure dependence of the moduli at lower pressures, we follow Goddard [10] and allow for new force-bearing contacts to be created as strain is applied to the grain packs. Although Goddard's ideas [10] have certainly received some attention [9,14,[17][18][19][20][21] in the granular packing literature, the present authors feel more attention to the details is appropriate, and this is the main contribution that is being made here.…”

“…We then follow Goddard [10], and present two mechanisms for how rattlers become jammed as a function of applied strain and show how to incorporate such mechanisms of gap closure into an analytical model for the elastic moduli.…”

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

“…The local extrema in the dependences of the harmonics amplitudes on the layer thickness in (29), (32) and (33). For this purpose, we remind that in reflection from a mechanically free surface of an elastically homogeneous medium 1) displacement waves of all frequencies are reflected without a phase shift;…”

“…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].…”