  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.

Soft Condensed Matter;Disordered Systems and Neural Networks;Materials Science