Bulk modulus along jamming transition lines of bidisperse granular packings

Abstract

We present three-dimensional discrete element method simulations of bidisperse granular packings to investigate their jamming densities ${\ensuremath{\phi}}_{J}$ and dimensionless bulk moduli $K$ as functions of the size ratio $\ensuremath{\delta}$ and the concentration of small particles ${X}_{\mathrm{S}}$. We determine the partial and total bulk moduli for packings near their jamming densities, including a second transition that occurs for sufficiently small $\ensuremath{\delta}$ and ${X}_{\mathrm{S}}$ when the system is compressed beyond its first jamming transition. While the first transition is sharp, exclusively with large-large contacts, the second is rather smooth, carried by small-large interactions at densities much higher than the monodisperse random packing baseline, ${\ensuremath{\phi}}_{J}^{\mathrm{mono}}\ensuremath{\approx}0.64$. When only nonrattlers are considered, all the effective transition densities are reduced, and the density of the second transition emerges rather close to the reduced baseline, ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\phi}}}_{J}^{\mathrm{mono}}\ensuremath{\approx}0.61$, due to its smooth nature. At size ratios $\ensuremath{\delta}\ensuremath{\le}0.22$ a concentration ${X}_{\mathrm{S}}^{*}$ divides the diagram---either with most small particles nonjammed or jammed jointly with large ones. For ${X}_{\mathrm{S}}<{X}_{\mathrm{S}}^{*}$, the modulus $K$ displays different behaviors at first and second jamming transitions. Along the second transition, $K$ rises relative to the values found at the first transition; however, is still small compared to $K$ at ${X}_{\mathrm{S}}^{*}$. Explicitly, for our smallest $\ensuremath{\delta}=0.15$, the discontinuous jump in $K$ as a function of ${X}_{\mathrm{S}}$ is obtained at ${X}_{\mathrm{S}}^{*}\ensuremath{\approx}0.21$ and coincides with the maximum jamming density where both particle species mix most efficiently. Our results will allow tuning or switching the bulk modulus $K$ or other properties, such as the wave speed, by choosing specific sizes and concentrations based on a better understanding of whether small particles contribute to the jammed structure or not, and how the micromechanical structure behaves at either transition.

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