Nonlinear Dynamics in Granular Columns

Abstract

Atomistic simulations of the vertical propagation of perturbations in deep gravitationally compacted granular columns characterized by the intergrain potential $V\left(\ensuremath{\delta}\right)\ensuremath{\propto}{\ensuremath{\delta}}^{n}$, with $n\ensuremath{\ge}2$ and $\ensuremath{\delta}$ the grain overlap, are shown to recover the results of elasticity theory for weak perturbations. For strong perturbations, the sound velocity ${c}_{\mathrm{strong}}\ensuremath{\rightarrow}{c}_{\mathrm{weak}}$ as $z\ensuremath{\rightarrow}\ensuremath{\infty}$ with the deviations from ${c}_{\mathrm{weak}}$ best expressed via a certain recursion relation in $z$. We predict that voids in real granular columns lead to $c\ensuremath{\propto}1\ensuremath{-}\ensuremath{\epsilon}$, when the void fraction $\ensuremath{\epsilon}$ is small, and show that the velocity power spectrum of a grain resembles that of a harmonic oscillator chain as $z\ensuremath{\rightarrow}\ensuremath{\infty}$.