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 ϕ_{J} and dimensionless bulk moduli K as functions of the size ratio δ and the concentration of small particles X_{S}. We determine the partial and total bulk moduli for packings near their jamming densities, including a second transition that occurs for sufficiently small δ and X_{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, ϕ_{J}^{mono}≈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, ϕ[over ̃]_{J}^{mono}≈0.61, due to its smooth nature. At size ratios δ≤0.22 a concentration X_{S}^{*} divides the diagram-either with most small particles nonjammed or jammed jointly with large ones. For X_{S}<X_{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_{S}^{*}. Explicitly, for our smallest δ=0.15, the discontinuous jump in K as a function of X_{S} is obtained at X_{S}^{*}≈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.