Elastic Moduli and Vibrational Modes in Jammed Particulate Packings

TL;DR

This study investigates how vibrational modes influence the elastic moduli of jammed particulate packings, revealing critical differences in bulk and shear responses near the jamming transition through modal decomposition analysis.

Contribution

It introduces a modal decomposition approach to understand the non-affine elastic response in jammed packings, highlighting distinct behaviors of bulk and shear moduli at the transition.

Findings

Bulk modulus remains finite at jamming transition.

Shear modulus approaches zero at jamming transition.

Vibrational density of states influences non-affine elastic response.

Abstract

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 $\varphi$. 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, $\varphi_{c}$, the vibrational density of states, $g(\omega)$, shows a plateau in the intermediate frequency regime above a characteristic frequency $\omega^\ast$. We illustrate that this unusual feature apparent in $g(\omega)$ is reflected in the behavior of $M_N$: As $\varphi \rightarrow \varphi_c$, where $\omega^\ast \rightarrow 0$, those modes for $\omega < \omega^\ast$ contribute less and less, while contributions from those for $\omega > \omega^\ast$ approach a constant value which results in $M_N$ to approach a critical value $M_{Nc}$, as $M_N-M_{Nc} \sim \omega^\ast$. At $\varphi_c$ 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.