Nonlinear elasticity and pressure-dependent wave speeds in granular media

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 p½ dependence of incremental elastic moduli on pressure p, a dependence observed in numerous experiments but found to be at odds with the p½ 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 p½pressure scaling at low pressure and both exhibit a high-pressure transition to p½ 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* = cμˆ∝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 μˆ 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.