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$^{\frac{1}{2}}$ dependence of incremental elastic moduli on pressure p, a dependence observed in numerous experiments but found to be at odds with the p$^{\frac{1}{3}}$ 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$^{\frac{1}{2}}$ pressure scaling at low pressure and both exhibit a high-pressure transition to p$^{\frac{1}{3}}$ scaling at a characteristic transition pressure p$^{\ast}$. For assemblages of nearly equal spheres, a non-hertzian contact model for mechanism (1) and percolation-type model for (2) yield estimates of p$^{\ast}$ of the form p$^{\ast}$ = c$\hat{\mu}\alpha ^{3}$. Here c is a non-dimensional coefficient depending only on granular-contact geometry, while $\alpha \ll $ 1 is a small parameter representing spherical imperfections and $\hat{\mu}$ 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 $\alpha $ = (h/R)$^{\frac{1}{2}}$ for mechanism (1) as opposed to $\alpha $ = 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.