Nonlinear Dynamics in Granular Columns

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