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Elastic Moduli and Vibrational Modes in Jammed Particulate Packings

When we elastically impose a homogeneous, affine deformation on amorphous solids, they also undergo an inhomogeneous, non-affine deformation, which can have a crucial impact on the overall elastic response. To correctly understand the elastic modulus $M$, it is therefore necessary to take into account not only the affine modulus $M_A$, but also the non-affine modulus $M_N$ that arises from the non-affine deformation. In the present work, we study the bulk ($M=K$) and shear ($M=G$) moduli in static jammed particulate packings over a range of packing fractions $φ$. One novelty of this work is to elucidate the contribution of each vibrational mode to the non-affine $M_N$ through a modal decomposition of the displacement and force fields. In the vicinity of the (un)jamming transition, $φ_{c}$, the vibrational density of states, $g(ω)$, shows a plateau in the intermediate frequency regime above a characteristic frequency $ω^\ast$. We illustrate that this unusual feature apparent in $g(ω)$ is reflected in the behavior of $M_N$: As $φ\rightarrow φ_c$, where $ω^\ast \rightarrow 0$, those modes for $ω< ω^\ast$ contribute less and less, while contributions from those for $ω> ω^\ast$ approach a constant value which results in $M_N$ to approach a critical value $M_{Nc}$, as $M_N-M_{Nc} \sim ω^\ast$. At $φ_c$ itself, the bulk modulus attains a finite value $K_c=K_{Ac}-K_{Nc} > 0$, such that $K_{Nc}$ has a value that remains below $K_{Ac}$. In contrast, for the critical shear modulus $G_c$, $G_{Nc}$ and $G_{Ac}$ approach the same value so that the total value becomes exactly zero, $G_c = G_{Ac}-G_{Nc} =0$. We explore what features of the configurational and vibrational properties cause such the distinction between $K$ and $G$, allowing us to validate analytical expressions for their critical values.